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Lemma 4.3. Let \( M \) be a left \( R \) -module, let \( E = {\operatorname{End}}_{R}^{\mathrm{{op}}}\left( M\right) \), let \( n > 0 \) , and let \( F = {\operatorname{End}}_{R}^{\mathrm{{op}}}\left( {M}^{n}\right) \) . If \( \xi : M \rightarrow M \) is an E-endomorphism, then \( {\xi }^{n} : {M}^{n} \rightarrow {M}^{... | Proof. Let \( {\iota }_{i} : M \rightarrow {M}^{n} \) and \( {\pi }_{j} : {M}^{n} \rightarrow M \) be the injections and projections. Let \( \eta \in F \) . As in the proof of 1.4, every \( \eta : {M}^{n} \rightarrow {M}^{n} \) is determined by a matrix \( \left( {\eta }_{ij}\right) \), where \( {\eta }_{ij} = {\pi }_{... | No |
Corollary 4.4. If \( D \) is a division ring, then the center of \( {M}_{n}\left( D\right) \) consists of all scalar matrices whose diagonal entry is in the center of \( D \) . | Proof. We prove this for \( {D}^{\mathrm{{op}}} \) . Let \( I \) denote the identity matrix. Let \( V \) be a left \( D \) -module with a basis \( {e}_{1},\ldots ,{e}_{n} \), so that \( {\operatorname{End}}_{D}\left( V\right) \cong {M}_{n}\left( {D}^{\mathrm{{op}}}\right) \) . Let \( E = {\operatorname{End}}_{D}^{\math... | Yes |
Corollary 4.5 (Burnside [1905]). Let \( K \) be an algebraically closed field, let \( V \) be a finite-dimensional vector space over \( K \), and let \( R \) be a subring of \( {\operatorname{End}}_{K}\left( V\right) \) that contains all scalar transformations \( v \mapsto {av} \) with \( a \in K \) . If \( V \) is a s... | Proof. Identify \( a \in K \) with the scalar transformation \( {aI} : v \mapsto {av} \), so that \( K \) becomes a subfield of \( {\operatorname{End}}_{K}\left( V\right) \) that is contained in the center of \( {\operatorname{End}}_{K}\left( V\right) \) (in fact, \( K \) is the center of \( {\operatorname{End}}_{K}\le... | Yes |
Theorem 4.6 (Jacobson Density Theorem). A ring \( R \) is left primitive if and only if it is isomorphic to a dense subring of \( {\operatorname{End}}_{D}\left( V\right) \) for some division ring \( D \) and right \( D \) -module \( V \) . | Proof. If \( S \) is a faithful simple left \( R \) -module, then \( D = {\operatorname{End}}_{R}^{\mathrm{{op}}}\left( S\right) \) is a division ring by 1.2, the canonical homomorphism \( \Phi : R \rightarrow {\operatorname{End}}_{D}\left( S\right) \) is injective since \( S \) is faithful, and \( \Phi \left( R\right)... | Yes |
Theorem 4.7. Let \( R \) be a left primitive ring and let \( S \) be a faithful simple left \( R \) -module, so that \( D = {\operatorname{End}}_{R}^{\text{op }}\left( S\right) \) is a division ring.\n\n(1) If \( R \) is left Artinian, then \( n = {\dim }_{D}S \) is finite and \( R \cong {M}_{n}\left( D\right) \) . | Proof. As in the proof of Theorem 4.6, the canonical homomorphism \( \Phi \) : \( R \rightarrow {\operatorname{End}}_{D}\left( S\right) \) is injective, and \( \Phi \left( R\right) \) is dense in \( {\operatorname{End}}_{D}\left( S\right) \) .\n\n(1). If a basis of \( S \) contains an infinite sequence \( {e}_{1},\ldot... | Yes |
In a ring \( R, J\left( R\right) \) is the intersection of all the annihilators of simple left \( R \) -modules; hence \( J\left( R\right) \) is a two-sided ideal of \( R \) . | Proof. If \( L \) is a maximal left ideal of \( R \), then \( S = {}_{R}R/L \) is a simple left \( R \) -module and \( \operatorname{Ann}\left( S\right) \subseteq L \) . Hence the intersection of all \( \operatorname{Ann}\left( S\right) \) is contained in \( J\left( R\right) \) . Conversely, let \( r \in J\left( R\righ... | Yes |
Lemma 5.2. If \( x \in R \), then \( x \in J\left( R\right) \) if and only if \( 1 + {tx} \) has a left inverse for every \( t \in R \) . | Proof. If \( x \notin J\left( R\right) \), then \( x \notin L \) for some maximal left ideal \( L, L + {Rx} = R \) , \( 1 = \ell + {rx} \) for some \( \ell \in L \) and \( r \in R \), and \( 1 - {rx} \in L \) has no left inverse, since all its left multiples are in \( L \) . Conversely, if some \( 1 + {tx} \) has no le... | Yes |
Proposition 5.3. In a ring \( R, J\left( R\right) \) is the largest two-sided ideal \( I \) of \( R \) such that \( 1 + x \) is a unit of \( R \) for all \( x \in I \) ; hence \( J\left( R\right) = J\left( {R}^{\mathrm{{op}}}\right) \) . | Proof. If \( x \in J\left( R\right) \), then, by 5.2, \( 1 + x \) has a left inverse \( y \), whence \( y = 1 - {yx} \) and \( y \) has a left inverse \( z \) . Since \( y \) already has a right inverse \( 1 + x \), it follows that \( 1 + x = z \), and \( y \) is a two-sided inverse of \( 1 + x \) . Thus \( J\left( R\r... | Yes |
Proposition 5.6. In a ring \( R, J\left( R\right) \) contains all nilpotent left or right ideals of \( R \) . If \( R \) is commutative, then \( J\left( R\right) \) contains all nilpotent elements of \( R \) . | Proof. Let \( N \) be a nilpotent left ideal and let \( S \) be a simple left \( R \) -module. If \( {NS} \neq 0 \), then \( {NS} = S \) and \( S = {NS} = {N}^{2}S = \cdots = {N}^{n}S = 0 \), a contradiction; therefore \( {NS} = 0 \) and \( N \subseteq \operatorname{Ann}\left( S\right) \) . Hence \( N \subseteq J\left(... | Yes |
Proposition 5.7 (Nakayama’s Lemma). Let \( M \) be a finitely generated left \( R \) -module. If \( J\left( R\right) M = M \), then \( M = 0 \) . | Proof. Assume \( M \neq 0 \) . Since \( M \) is finitely generated, the union of a chain \( {\left( {N}_{i}\right) }_{i \in I} \) of proper submodules of \( M \) is a proper submodule of \( M \) : otherwise, the finitely many generators of \( M \) all belong to some \( {N}_{i} \), and then \( {N}_{i} = M \) . By Zorn’s... | Yes |
Proposition 5.8 (Nakayama’s Lemma). Let \( N \) be a submodule of a finitely generated left \( R \) -module \( M \) . If \( N + J\left( R\right) M = M \), then \( N = M \) . | Proof. First, \( M/N \) is finitely generated. If \( N + J\left( R\right) M = M \), then \( J\left( R\right) \left( {M/N}\right) = M/N \) and \( M/N = 0 \), by 5.7. \( ▱ \) | Yes |
Theorem 6.1. A ring \( R \) is semisimple if and only if \( R \) is left Artinian and \( J\left( R\right) = 0 \), if and only if \( R \) is left Artinian and has no nonzero nilpotent ideal. | Proof. The last two conditions are equivalent, by 6.2 below.\n\nBy 3.5,3.3, a semisimple ring \( R \) is left Artinian, and is isomorphic to a direct product \( {M}_{{n}_{1}}\left( {D}_{1}\right) \times \cdots \times {M}_{{n}_{s}}\left( {D}_{s}\right) \) of finitely many matrix rings over division rings \( {D}_{1},\ldo... | No |
Lemma 6.2. If \( R \) is left Artinian, then \( J\left( R\right) \) is nilpotent, and is the greatest nilpotent left ideal of \( R \) and the greatest nilpotent right ideal of \( R \) . | Proof. Let \( J = J\left( R\right) \) . Since \( R \) is left Artinian, the descending sequence \( J \supseteq {J}^{2} \supseteq \cdots \supseteq {J}^{n} \supseteq {J}^{n + 1} \supseteq \cdots \) terminates at some \( {J}^{m}\left( {{J}^{n} = {J}^{m}}\right. \) for all \( n \geqq m) \) . Suppose that \( {J}^{m} \neq 0 ... | Yes |
Proposition 6.3. If \( R \) is left Artinian, then a left \( R \) -module \( M \) is semisimple if and only if \( J\left( R\right) M = 0 \) . | Proof. Let \( J = J\left( R\right) \) . We have \( {JS} = 0 \) for every simple \( R \) -module \( S \), since \( J \subseteq \operatorname{Ann}\left( S\right) \) ; hence \( {JM} = 0 \) whenever \( M \) is semisimple. Conversely, assume that \( {JM} = 0 \) . Then \( M \) is a left \( R/J \) -module, in which \( \left( ... | Yes |
Theorem 6.4 (Hopkins-Levitzki). If \( R \) is left Artinian, then for a left \( R \) -module \( M \) the following properties are equivalent: (i) \( M \) is Noetherian; (ii) \( M \) is Artinian; (iii) \( M \) is of finite length. | Proof. If \( M \) is semisimple, then (i),(ii), and (iii) are equivalent, since a Noetherian or Artinian module cannot be the direct sum of infinitely many simple submodules.\n\nIn general, let \( J = J\left( R\right) \). Let \( M \) be Noetherian (or Artinian). By \( {6.2},{J}^{n} = 0 \) for some \( n > 0 \), which yi... | Yes |
Proposition 7.1. Every finite-dimensional representation is a direct sum of irreducible representations. | Proof. This is shown by induction on the dimension of \( V \) : if \( \rho \) is not irreducible, then \( \rho = {\rho }_{1} \oplus {\rho }_{2} \), where \( V = {V}_{1} \oplus {V}_{2} \) and \( {V}_{1},{V}_{2} \) have lower dimension than \( V \) . \( ▱ \) | Yes |
Proposition 7.2. Every multiplicative homomorphism of a group \( G \) into a \( K \) -algebra \( A \) extends uniquely to an algebra homomorphism of \( K\left\lbrack G\right\rbrack \) into \( A \) . | Proof. Let \( \varphi : G \rightarrow A \) be a multiplicative homomorphism (meaning \( \varphi \left( {gh}\right) = \varphi \left( g\right) \varphi \left( h\right) \) for all \( g, h \in G \), and \( \varphi \left( 1\right) = 1 \) ). Since \( G \) is a basis of \( K\left\lbrack G\right\rbrack ,\varphi \) extends uniqu... | Yes |
There is a one-to-one correspondence between representations of a group \( G \) over a field \( K \) and \( K\left\lbrack G\right\rbrack \) -modules. Two representations are equivalent if and only if the corresponding modules are isomorphic. | Proof. By 7.2, a representation \( \rho : G \rightarrow {GL}\left( V\right) \subseteq {\operatorname{End}}_{K}\left( V\right) \) extends uniquely to an algebra homomorphism \( \rho : K\left\lbrack G\right\rbrack \rightarrow {\operatorname{End}}_{K}\left( V\right) \) that makes \( V \) a \( K\left\lbrack G\right\rbrack ... | Yes |
Theorem 7.4 (Maschke [1898]). Let \( G \) be a finite group and let \( K \) be a field. If \( K \) has characteristic 0, or if \( K \) has characteristic \( p \neq 0 \) and \( p \) does not divide the order of \( G \), then \( K\left\lbrack G\right\rbrack \) is semisimple. | Proof. We show that every submodule \( W \) of a \( G \) -module \( V \) is a direct summand of \( V \) . We already have \( V = W \oplus {W}^{\prime } \) (as a vector space) for some subspace \( {W}^{\prime } \) . The projection \( \pi : V \rightarrow W \) is a linear transformation and is the identity on \( W \) . De... | Yes |
Corollary 7.5. Let \( G \) be a finite group and let \( K \) be a field whose characteristic does not divide the order of \( G \) .\n\n(1) Up to isomorphism, there are only finitely many simple \( G \) -modules \( {S}_{1},\ldots ,{S}_{s} \) , and they all have finite dimension over \( K \) .\n\n(2) Every G-module is is... | Proof. That there are only finitely many simple \( G \) -modules follows from 3.6. By 1.3, every simple \( G \) -module is isomorphic to a minimal left ideal of \( K\left\lbrack G\right\rbrack \) , and has finite dimension over \( K \), like \( K\left\lbrack G\right\rbrack \) . Then (2) follows from 3.7, and (3) follow... | No |
Proposition 7.6. Let \( G \) be a finite group and let \( K \) be an algebraically closed field whose characteristic does not divide the order of \( G \) . Let the nonisomorphic simple \( G \) -modules have dimensions \( {d}_{1},\ldots ,{d}_{s} \) over \( K \) . The simple components of \( K\left\lbrack G\right\rbrack ... | Proof. By 7.4, \( K\left\lbrack G\right\rbrack \) is semisimple. The simple components \( {R}_{1},\ldots ,{R}_{r} \) of \( K\left\lbrack G\right\rbrack \) are simple left Artinian rings \( {R}_{i} \cong {M}_{{n}_{i}}\left( {D}_{i}\right) \), where \( {D}_{1},\ldots ,{D}_{r} \) are division rings, as well as two-sided i... | Yes |
Lemma 7.7. Let \( D \) be a division ring that has finite dimension over a central subfield \( K \) . If \( K \) is algebraically closed, then \( D = K \) . | Proof. If \( \alpha \in D \), then \( K \) and \( \alpha \) generate a commutative subring \( K\left\lbrack \alpha \right\rbrack \) of \( D \) , \( \alpha \) is algebraic over \( K \) since \( K\left\lbrack \alpha \right\rbrack \) has finite dimension over \( K \), and \( \alpha \in K \) since \( K \) is algebraically ... | Yes |
Theorem 7.9. Let \( G \) be a finite group with \( s \) conjugacy classes and let \( K \) be an algebraically closed field whose characteristic does not divide the order of \( G \) . Up to equivalence, \( G \) has \( s \) distinct irreducible representations over \( K \) . | Proof. We look at the center \( Z\left( {K\left\lbrack G\right\rbrack }\right) \) of \( K\left\lbrack G\right\rbrack \) . By 7.6, \( K\left\lbrack G\right\rbrack \cong {R}_{1} \times \cdots \times \) \( {R}_{s} \), where \( s \) is now the number of distinct irreducible representations of \( G \) and \( {R}_{i} \cong {... | No |
Proposition 7.10. Let \( G \) be a finite group and let \( K \) be an algebraically closed field whose characteristic does not divide the order of \( G \) . Let \( \rho : G \rightarrow {\operatorname{End}}_{K}\left( S\right) \) be an irreducible representation and let \( c \) be the sum of a conjugacy class. Then \( \r... | Proof. First, \( S \) is a simple \( K\left\lbrack G\right\rbrack \) -module. Let \( E = {\operatorname{End}}_{K\left\lbrack G\right\rbrack }^{\text{op }}\left( S\right) \) . Since \( K \) is central in \( K\left\lbrack G\right\rbrack \), the scalar linear transformations \( ╏ : x \mapsto {\lambda x} \) are \( K\left\l... | Yes |
Proposition 8.2. If \( K \) is an algebraically closed field whose characteristic does not divide \( \left| G\right| \), then the regular character is \( {\chi }_{r} = \mathop{\sum }\limits_{i}{d}_{i}{\chi }_{i} \) . | Proof. By 7.6, \( {R}_{i} \cong {M}_{{d}_{i}}\left( K\right) \cong {S}_{i}^{{d}_{i}} \) ; hence \( K\left\lbrack G\right\rbrack \cong {S}_{1}^{{d}_{1}} \oplus \cdots \oplus {S}_{s}^{{d}_{s}} \), as a \( G \) -module, and \( {\chi }_{r} = \mathop{\sum }\limits_{i}{d}_{i}{\chi }_{i} \) . \( ▱ \) | Yes |
Lemma 8.3. If the characteristic of \( K \) does not divide \( \left| G\right| \), then:\n\n(1) \( {\chi }_{i}\left( x\right) = 0 \) when \( x \in {R}_{j} \) and \( j \neq i \) ;\n\n(2) \( {\chi }_{i}\left( {e}_{j}\right) = 0 \) if \( j \neq i \) and \( {\chi }_{i}\left( {e}_{i}\right) = {\chi }_{i}\left( 1\right) = {d... | Proof. If \( x \in {R}_{j} \) and \( j \neq i \), then \( {\rho }_{i}\left( x\right) \left( s\right) = {xs} = 0 \) for all \( s \in {S}_{i} \) , \( {\rho }_{i}\left( x\right) = 0 \), and \( {\chi }_{i}\left( x\right) = \operatorname{Tr}{\rho }_{i}\left( x\right) = 0 \) . In particular, \( {\chi }_{i}\left( {e}_{j}\righ... | Yes |
Theorem 8.4. If \( G \) is finite and \( K \) has characteristic 0 :\n\n(1) the irreducible characters of \( G \) are linearly independent over \( K \) ;\n\n(2) every character of \( G \) can be written uniquely as a linear combination of irreducible characters with nonnegative integer coefficients;\n\n(3) two finite-d... | Proof. If \( {a}_{i} \in K \) and \( \mathop{\sum }\limits_{i}{a}_{i}{\chi }_{i} = 0 \), then \( {d}_{i}{a}_{i} = \mathop{\sum }\limits_{j}{a}_{j}{\chi }_{j}\left( {e}_{i}\right) = 0 \) for all \( i \) ; since \( K \) has characteristic 0 this implies \( {a}_{i} = 0 \) for all \( i \) .\n\nLet \( \chi \) be the charact... | Yes |
Theorem 8.5. If \( G \) is a finite group and \( K \) is an algebraically closed field whose characteristic does not divide \( \left| G\right| \), then \( {d}_{i} \neq 0 \) in \( K \) and\n\n\[ \mathop{\sum }\limits_{{g \in G}}{\chi }_{i}\left( g\right) {\chi }_{j}\left( {g}^{-1}\right) = \left\{ \begin{array}{ll} \lef... | Proof. Let \( {e}_{ig} \) be the \( g \) coordinate of \( {e}_{i} \) in \( K\left\lbrack G\right\rbrack \), so that \( {e}_{i} = \mathop{\sum }\limits_{{g \in G}}{e}_{ig}g \) . Since \( {\chi }_{r}\left( 1\right) = \left| G\right| \) and \( {\chi }_{r}\left( g\right) = 0 \) for all \( g \neq 1 \) we have\n\n\[ {\chi }_... | Yes |
Proposition 9.1. Let \( G \) be a finite group and let \( \chi \) be the character of a complex representation \( \rho : G \rightarrow {GL}\left( V\right) \) of dimension \( d \) . For every \( g \in G \) :\n\n(1) \( \rho \left( g\right) \) is diagonalizable;\n\n(2) \( \chi \left( g\right) \) is a sum of \( d \) roots ... | Proof. (1). Let \( H = \langle g\rangle \) . By 7.8 the representation \( {\rho }_{\mid H} : H \rightarrow {GL}\left( V\right) \) is a direct sum of representations of dimension 1. Hence \( V \), as an \( H \) -module, is a direct sum of submodules of dimension 1 over \( \mathbb{C} \), and has a basis \( {e}_{1},\ldots... | Yes |
Lemma 9.2. \( {c}_{ij} \) is integral over \( \mathbb{Z} \) . | Proof. First, \( {c}_{j}{c}_{k} = \left( {\mathop{\sum }\limits_{{{h}^{\prime } \in {C}_{j}}}{h}^{\prime }}\right) \left( {\mathop{\sum }\limits_{{{h}^{\prime \prime } \in {C}_{k}}}{h}^{\prime \prime }}\right) = \mathop{\sum }\limits_{{g \in G}}{n}_{g}g \), where \( {n}_{g} \) is the number of ordered pairs \( \left( {... | Yes |
Proposition 9.3. If \( {d}_{i} \) and \( \left| {C}_{j}\right| \) are relatively prime, then either \( \left| {{\chi }_{i}\left( g\right) }\right| = {d}_{i} \) for all \( g \in {C}_{j} \), or \( {\chi }_{i}\left( g\right) = 0 \) for all \( g \in {C}_{j} \) . | Proof. Assume \( \left| {{\chi }_{i}\left( g\right) }\right| < {d}_{i} \), where \( g \in {C}_{j} \) . Let \( \alpha = {\chi }_{i}\left( g\right) /{d}_{i} = {c}_{ij}/\left| {C}_{j}\right| \) . Then \( \left| \alpha \right| < 1 \) . Also, \( u{d}_{i} + v\left| {C}_{j}\right| = 1 \) for some \( u, v \in \mathbb{Z} \) ; h... | Yes |
Theorem 9.4 (Burnside). Let \( p \) and \( q \) be prime numbers. Every group of order \( {p}^{m}{q}^{n} \) is solvable. | Proof. It is enough to show that simple groups of order \( {p}^{m}{q}^{n} \) are abelian.\n\nAssume that \( G \) is a simple nonabelian group of order \( {p}^{m}{q}^{n} \) . Since \( p \) -groups are solvable we may assume that \( p \neq q \) and that \( m, n > 0 \) . Number \( {\chi }_{1},\ldots ,{\chi }_{s} \) so tha... | Yes |
Lemma 1.1. If \( 0 \rightarrow A\overset{\mu }{ \rightarrow }B\overset{\varphi }{ \rightarrow }C \) is exact, then every homomorphism \( \psi \) such that \( \varphi \circ \psi = 0 \) factors uniquely through \( \mu \): | Proof. We have \( \operatorname{Im}\psi \subseteq \operatorname{Ker}\varphi = \operatorname{Im}\mu \) . Since \( \mu \) is injective there is a unique mapping \( \chi : M \rightarrow A \) such that \( \psi \left( x\right) = \mu \left( {\chi \left( x\right) }\right) \) for all \( x \in M \) . Then \( \chi \) is a module... | Yes |
Lemma 1.2. If \( A\overset{\varphi }{ \rightarrow }B\overset{\sigma }{ \rightarrow }C \rightarrow 0 \) is exact, then every homomorphism \( \psi \) such that \( \psi \circ \varphi = 0 \) factors uniquely through \( \sigma \) : | Proof. This follows from Theorem VIII.2.5, since Ker \( \sigma = \operatorname{Im}\varphi \subseteq \operatorname{Ker}\psi \) . \( ▱ \) | Yes |
Lemma 1.3 (Short Five Lemma). In a commutative diagram with exact rows,\n\n\n\nif \( \alpha \) and \( \gamma \) are isomorphisms, then so is \( \beta \) . | Proof. Assume that \( \beta \left( b\right) = 0 \) . Then \( \gamma \left( {\rho \left( b\right) }\right) = {\rho }^{\prime }\left( {\beta \left( b\right) }\right) = 0 \) and \( \rho \left( b\right) = 0 \) . By exactness, \( b = \mu \left( a\right) \) for some \( a \in A \) . Then \( {\mu }^{\prime }\left( {\alpha \lef... | Yes |
Proposition 1.5. For a short exact sequence \( 0 \rightarrow A\overset{\mu }{ \rightarrow }B\overset{\rho }{ \rightarrow }C \rightarrow 0 \) the following conditions are equivalent:\n\n(1) \( \mu \) splits \( \left( {\sigma \circ \mu = {1}_{A}}\right. \) for some homomorphism \( \sigma : B \rightarrow A \) );\n\n(2) \(... | Proof. (1) implies (3). Assume that \( \sigma \circ \mu = {1}_{A} \) . There is a homomorphism \( \theta : B \rightarrow A \oplus C \) such that \( {\pi }^{\prime } \circ \theta = \sigma \) and \( \pi \circ \theta = \rho \) (where \( {\pi }^{\prime } \) : \( A \oplus C \rightarrow A \) is the projection), namely \( {\t... | Yes |
Proposition 2.1. For every pair of homomorphisms \( \alpha : A \rightarrow C \) and \( \beta \) : \( B \rightarrow C \) of left \( R \) -modules, there exists a pullback \( \alpha \circ {\beta }^{\prime } = \beta \circ {\alpha }^{\prime } \), and it is unique up to isomorphism. | Proof. Uniqueness follows from the universal property. Let \( \alpha \circ {\beta }^{\prime } = \beta \circ {\alpha }^{\prime } \) and \( \alpha \circ {\beta }^{\prime \prime } = \beta \circ {\alpha }^{\prime \prime } \) be pullbacks. There exist homomorphisms \( \theta \) and \( \zeta \) such that \( {\beta }^{\prime ... | Yes |
Proposition 2.2. Given homomorphisms \( \alpha : A \rightarrow C \) and \( \beta : B \rightarrow C \) of left \( R \) -modules, let \( P = \{ \left( {a, b}\right) \in A \oplus B \mid \alpha \left( a\right) = \beta \left( b\right) \} ,{\alpha }^{\prime } : P \rightarrow B \) , \( \left( {a, b}\right) \mapsto b \), and \... | Proof. First, \( \alpha \circ {\beta }^{\prime } = \beta \circ {\alpha }^{\prime } \), by the choice of \( P \) . Assume \( \alpha \circ \varphi = \beta \circ \psi \) , where \( \varphi : M \rightarrow A \) and \( \psi : M \rightarrow B \) are module homomorphisms. Then \( \left( {\varphi \left( m\right) ,\psi \left( m... | Yes |
Proposition 2.5. For every pair of homomorphisms \( \alpha : C \rightarrow A \) and \( \beta \) : \( C \rightarrow B \) of left \( R \) -modules, there exists a pushout \( {\beta }^{\prime } \circ \alpha = {\alpha }^{\prime } \circ \beta \), and it is unique up to isomorphism. | Uniqueness follows from the universal property, and existence from a construction: | No |
Proposition 2.6. Given homomorphisms \( \alpha : C \rightarrow A \) and \( \beta : C \rightarrow B \) of left \( R \) -modules, let \( K = \{ \left( {\alpha \left( c\right) , - \beta \left( c\right) }\right) \in A \oplus B \mid c \in C\}, P = \left( {A \oplus B}\right) /K \) , \( {\alpha }^{\prime } = \pi \circ \kappa ... | Proof. First, \( \operatorname{Ker}\pi = K = \operatorname{Im}\left( {\iota \circ \alpha - \kappa \circ \beta }\right) \) ; hence \( {\beta }^{\prime } \circ \alpha = \pi \circ \iota \circ \alpha = \) \( \pi \circ \kappa \circ \beta = {\alpha }^{\prime } \circ \beta \) . Assume that \( \varphi \circ \alpha = \chi \circ... | Yes |
Proposition 3.1. Every free module is projective. | Proof. Let \( \varphi : F \rightarrow N \) and \( \rho : M \rightarrow N \) be homomorphisms and let \( {\left( {e}_{i}\right) }_{i \in I} \) be a basis of \( F \) . If \( \rho \) is surjective, there is for every \( i \in I \) some \( {m}_{i} \in M \) such that \( \varphi \left( {e}_{i}\right) = \rho \left( {m}_{i}\ri... | Yes |
Proposition 3.2. A left \( R \) -module \( P \) is projective if and only if every epimorphism \( M \rightarrow P \) splits, if and only if every short exact sequence \( 0 \rightarrow A \rightarrow \) \( B \rightarrow P \rightarrow 0 \) splits. | Proof. By 1.5, \( 0 \rightarrow A \rightarrow B \rightarrow P \rightarrow 0 \) splits if and only if \( B \rightarrow P \) splits. Assume that every epimorphism \( M \rightarrow P \) splits. Let \( \varphi : P \rightarrow N \) and \( \rho : M \rightarrow N \) be homomorphisms, with \( \rho \) surjective. In the pullbac... | No |
Corollary 3.3. A ring \( R \) is semisimple if and only if every short exact sequence of left \( R \) -modules splits, if and only if every left \( R \) -module is projective. | Proof. A left \( R \) -module \( B \) is semisimple if and only if every submodule of \( B \) is a direct summand, if and only if every short exact sequence \( 0 \rightarrow A \rightarrow B \rightarrow \) \( C \rightarrow 0 \) splits. \( ▱ \) | No |
Corollary 3.6. A module is projective if and only if it is isomorphic to a direct summand of a free module. | Proof. For every module \( P \) there is by VIII.4.6 an epimorphism \( \rho : F \rightarrow P \) where \( F \) is free; if \( P \) is projective, then the exact sequence \( 0 \rightarrow \operatorname{Ker}\varphi \rightarrow \) \( F \rightarrow P \rightarrow 0 \) splits, and \( P \) is isomorphic to a direct summand of... | Yes |
Corollary 3.7. If \( R \) is a PID, then an \( R \) -module is projective if and only if it is free. | Proof. Every submodule of a free \( R \) -module is free, by Theorem VIII.6.1. \( ▱ \) | No |
Proposition 4.1. For a left \( R \) -module \( J \) the following conditions are equivalent:\n\n(1) \( J \) is injective;\n\n(2) every monomorphism \( J \rightarrow M \) splits;\n\n(3) every short exact sequence \( 0 \rightarrow J \rightarrow B \rightarrow C \rightarrow 0 \) splits;\n\n(4) \( J \) is a direct summand o... | Proof. We prove that (2) implies (1); the other implications are clear. Assume (2). Let \( \varphi : M \rightarrow J \) and \( \mu : M \rightarrow N \) be homomorphisms, with \( \mu \) injective. In the pushout \( {\varphi }^{\prime } \circ \mu = {\mu }^{\prime } \circ \varphi ,{\mu }^{\prime } \) is injective, by 2.7.... | No |
Proposition 4.5 (Baer’s Criterion). For a left R-module \( J \) the following conditions are equivalent:\n\n(1) \( J \) is injective;\n\n(2) every module homomorphism of a left ideal of \( R \) into \( J \) can be extended to \( {}_{R}R \) ;\n\n(3) for every module homomorphism \( \varphi \) of a left ideal \( L \) of ... | Proof. (1) implies (2), and readers will happily show that (2) and (3) are equivalent. We show that (2) implies (1): every module homomorphism \( \varphi \) : \( M \rightarrow J \) extends through every monomorphism \( \mu : M \rightarrow N \) . Then \( M \) is isomorphic to a submodule of \( N \) ; we may assume that ... | No |
Proposition 4.6. If \( R \) is a domain, then every injective \( R \) -module is divisible. | Proof. Let \( J \) be injective. Let \( a \in J \) and \( 0 \neq r \in R \) . Since \( R \) is a domain, every element of \( {Rr} \) can be written in the form \( {tr} \) for some unique \( t \in R \) . Hence \( \varphi : {tr} \mapsto {ta} \) is a module homomorphism of \( {Rr} \) into \( J \) . By 4.5 there exists \( ... | Yes |
Proposition 4.7. If \( R \) is a PID, then an \( R \) -module is injective if and only if it is divisible. | Proof. Let \( M \) be a divisible \( R \) -module. Let \( {Rr} \) be any (left) ideal of \( R \) and let \( \varphi : {Rr} \rightarrow M \) be a module homomorphism. If \( r = 0 \), then \( \varphi \left( s\right) = {s0} \) for all \( s \in {Rr} \) . Otherwise \( \varphi \left( r\right) = {rm} \) for some \( m \in M \)... | No |
Proposition 4.8. The group \( {\mathbb{Z}}_{{p}^{\infty }} \) is the union of cyclic subgroups \( {C}_{1} \subseteq {C}_{2} \subseteq \) \( \cdots \subseteq {C}_{n} \subseteq \cdots \) of orders \( p,{p}^{2},\ldots ,{p}^{n},\ldots \) and is a divisible abelian group. | Proof. First we find a model of \( {\mathbb{Z}}_{{p}^{\infty }} \) . Let \( U \) be the multiplicative group of all complex numbers of modulus 1 . Let \( {\alpha }_{n} = {e}^{{2i\pi }/{p}^{n}} \in U \) . Then \( {\alpha }_{1}^{p} = 1 \) and \( {\alpha }_{n}^{p} = {\alpha }_{n - 1} \) for all \( n > 1 \) . Hence there i... | Yes |
Proposition 4.10. Every abelian group can be embedded into a divisible abelian group. | Proof. For every abelian group \( A \) there is an epimorphism \( F \rightarrow A \) where \( F \) is a free abelian group. Now, \( F \) is a direct sum of copies of \( \mathbb{Z} \) and can be embedded into a direct sum \( D \) of copies of \( \mathbb{Q} \) . Then \( D \) is divisible, like \( \mathbb{Q} \) . By 2.7, ... | Yes |
Theorem 4.12. A ring \( R \) is left Noetherian if and only if every direct sum of injective left \( R \) -modules is injective. | Proof. Assume that every direct sum of injective left \( R \) -modules is injective, and let \( {L}_{1} \subseteq {L}_{2} \subseteq \cdots \subseteq {L}_{n} \subseteq \cdots \) be an ascending sequence of left ideals of \( R \) . Then \( L = \mathop{\bigcup }\limits_{{n > 0}}{L}_{n} \) is a left ideal of \( R \) . By 4... | Yes |
Proposition 5.2. If \( \mu \) is an essential monomorphism, and \( \varphi \circ \mu \) is injective, then \( \varphi \) is injective. | Proof. If \( \varphi \circ \mu \) is injective, then \( \operatorname{Ker}\varphi \cap \operatorname{Im}\mu = 0 \) ; hence \( \operatorname{Ker}\varphi = 0 \) . | Yes |
Proposition 5.3. If \( \mu : A \rightarrow B \) and \( v : B \rightarrow C \) are monomorphisms, then \( v \circ \mu \) is essential if and only if \( \mu \) and \( v \) are essential. | This follows from Proposition 5.1; the details make nifty exercises. | No |
Proposition 5.4. A left \( R \) -module \( J \) is injective if and only if \( J \) has no proper essential extension \( J \subsetneqq M \), if and only if every essential monomorphism \( J \rightarrow M \) is an isomorphism. | Proof. Let \( J \) be injective. If \( J \subseteq M \), then \( J \) is a direct summand of \( M \) , \( M = J \oplus N \) ; then \( N \cap J = 0 \) ; if \( J \) is essential in \( M \), then \( N = 0 \) and \( J = M \) . If in turn \( J \) has no proper essential extension, and \( \mu : J \rightarrow M \) is an essen... | Yes |
Proposition 5.5. Let \( \mu : M \rightarrow N \) and \( v : M \rightarrow J \) be monomorphisms. If \( \mu \) is essential and \( J \) is injective, then \( v = \kappa \circ \mu \) for some monomorphism \( \kappa : N \rightarrow J \) . | Proof. Since \( J \) is injective, there exists a homomorphism \( \kappa : N \rightarrow J \) such that \( v = \kappa \circ \mu \), which is injective by 5.2. \( ▱ \) | No |
Theorem 5.6. Every left R-module \( M \) is an essential submodule of an injective \( R \) -module, which is unique up to isomorphism. | Proof. By Theorem 4.11, \( M \) is a submodule of an injective module \( K \) . Let \( \mathcal{S} \) be the set of all submodules \( M \subseteq S \subseteq K \) of \( K \) in which \( M \) is essential (for instance, \( M \) itself). If \( {\left( {S}_{i}\right) }_{i \in I} \) is a chain in \( \mathcal{S} \), then \(... | Yes |
A ring \( R \) is left hereditary if and only if every left ideal of \( R \) is projective, and then every submodule of a free left R-module is isomorphic to a direct sum of left ideals of \( R \) . | Proof. If \( R \) is left hereditary, then every left ideal of \( R \) is projective (as an \( R \) -module), since \( {}_{R}R \) is projective. Conversely, assume that every left ideal of \( R \) is projective. We show that every submodule of a free module is isomorphic to a direct sum of left ideals of \( R \) ; then... | Yes |
Lemma 6.2. (1) A module \( P \) is projective if and only if every module homomorphism \( P \rightarrow B \) factors through every epimorphism \( J \rightarrow B \) in which \( J \) is injective. | Proof. We prove (1) and let readers reverse arrows to prove (2). Assume that every module homomorphism \( P \rightarrow B \) factors through every epimorphism \( J \rightarrow B \) in which \( J \) is injective. Let \( \varphi : P \rightarrow B \) be a homomorphism and let \( \sigma : A \rightarrow B \) be any epimorph... | No |
Proposition 6.3. A ring \( R \) is left hereditary if and only if every quotient of an injective left \( R \) -module is injective. | Proof. Assume that \( R \) is left hereditary. Let \( J \) be an injective module and let \( \sigma : J \rightarrow B \) be an epimorphism. To prove that \( B \) is injective we use 6.2 and show that every homomorphism \( \varphi : A \rightarrow B \) factors through every monomorphism \( \mu : A \rightarrow P \) in whi... | Yes |
Proposition 6.4. A nonzero fractional ideal of a domain \( R \) is invertible if and only if it is projective as an \( R \)-module. | Proof. Let \( \mathfrak{A} \neq 0 \) be a fractional ideal of \( R \) . Let \( \mathfrak{{AB}} = R \) for some fractional ideal \( \mathfrak{B} \) . As in the proof of VII.5.2, \( 1 = {a}_{1}{b}_{1} + \cdots + {a}_{n}{b}_{n} \) for some \( {a}_{1},\ldots ,{a}_{n} \in \) \( \mathfrak{A} \) and \( {b}_{1},\ldots ,{b}_{n}... | Yes |
Proposition 6.6. A domain \( R \) is a Dedekind domain if and only if every divisible \( R \) -module is injective. | Proof. Injective modules are divisible, by 4.6, and quotients of divisible modules are divisible. If divisible \( R \) -modules are injective, then quotients of injective \( R \) -module are injective, \( R \) is hereditary by 6.3, and \( R \) is Dedekind by 6.5 .\n\nConversely, let \( R \) be a Dedekind domain and let... | Yes |
Proposition 1.1. If \( \varphi : A \rightarrow B \) is a homomorphism of left \( R \) -modules, then \( {\varphi }_{ * } = {\operatorname{Hom}}_{R}\left( {M,\varphi }\right) : {\operatorname{Hom}}_{R}\left( {M, A}\right) \rightarrow {\operatorname{Hom}}_{R}\left( {M, B}\right) \) is a homomorphism of abelian groups. Mo... | (1) if \( \varphi \) is the identity on \( A \), then \( {\varphi }_{ * } \) is the identity on \( {\operatorname{Hom}}_{R}\left( {M, A}\right) \) ;\n\n(2) \( {\left( \psi \circ \varphi \right) }_{ * } = {\psi }_{ * } \circ {\varphi }_{ * } \) whenever \( \varphi : A \rightarrow B \) and \( \psi : B \rightarrow C \) ;\... | Yes |
Proposition 1.3. For every \( \varphi : A \rightarrow B \) and \( \psi : M \rightarrow N \) the following square commutes:\n\n\[ \n{\operatorname{Hom}}_{R}\left( {M, A}\right) \xrightarrow[]{{\operatorname{Hom}}_{R}\left( {M,\varphi }\right) }{\operatorname{Hom}}_{R}\left( {M, B}\right) \]\n\n\[ \n{\operatorname{Hom}}_... | Proof. For every \( \alpha : N \rightarrow A,{\varphi }_{ * }\left( {{\psi }^{ * }\left( \alpha \right) }\right) = \varphi \circ \alpha \circ \psi = {\psi }^{ * }\left( {{\varphi }_{ * }\left( \alpha \right) }\right) \) . \( ▱ \) | Yes |
Proposition 1.4. If \( M \) is an \( R \) -S-bimodule and \( A \) is an \( R \) -T-bimodule, then \( {\operatorname{Hom}}_{R}\left( {M, A}\right) \) is an \( S \) - \( T \) -bimodule, in which\n\n\[ \left( {s\alpha }\right) \left( x\right) = \alpha \left( {xs}\right) \text{ and }\left( {\alpha t}\right) \left( x\right)... | Proof. In the above, \( {s\alpha } \) and \( {\alpha t} \) are homomorphisms of left \( R \) -modules, since \( M \) and \( A \) are bimodules. Moreover, \( s\left( {\alpha + \beta }\right) = {s\alpha } + {s\beta } \), and\n\n\[ s\left( {{s}^{\prime }\alpha }\right) \left( x\right) = \left( {{s}^{\prime }\alpha }\right... | Yes |
Proposition 1.6. If \( M \) is an \( R \) -S-bimodule and \( \varphi \) is a homomorphism of \( R \) -T-bimodules, then \( {\operatorname{Hom}}_{R}\left( {M,\varphi }\right) \) is a homomorphism of \( S \) - \( T \) -bimodules. If \( A \) is an \( R \) - \( T \) -bimodule and \( \psi \) is a homomorphism of \( R \) - \... | The proof is an exercise. In Proposition 1.6, \( {\operatorname{Hom}}_{R}\left( {M, - }\right) \) is now a functor from \( R \) - \( T \) -bimodules to \( S \) - \( T \) -bimodules, and \( {\operatorname{Hom}}_{R}\left( {-, A}\right) \) is a contravariant functor from \( R \) - \( S \) -bimodules to \( S \) - \( T \) -... | No |
Proposition 2.1 (Left Exactness). If \( 0 \rightarrow A\overset{\mu }{ \rightarrow }B\overset{\rho }{ \rightarrow }C \) is exact, then\n\n\( 0 \rightarrow {\operatorname{Hom}}_{R}\left( {M, A}\right) \xrightarrow[]{{\operatorname{Hom}}_{R}\left( {M,\mu }\right) }{\operatorname{Hom}}_{R}\left( {M, B}\right) \xrightarrow... | Proof. If \( {\mu }_{ * }\left( \alpha \right) = 0 \), then \( \mu \circ \alpha = 0,\operatorname{Im}\alpha \subseteq \operatorname{Ker}\mu = 0 \), and \( \alpha = 0 \) . Similarly, \( {\rho }_{ * }\left( {{\mu }_{ * }\left( \alpha \right) }\right) = \rho \circ \mu \circ \alpha = 0 \) . Conversely, if \( {\rho }_{ * }\... | Yes |
Proposition 2.5. There is an isomorphism\n\n\[ \n{\operatorname{Hom}}_{R}\left( {M,\mathop{\prod }\limits_{{i \in I}}{A}_{i}}\right) \cong \mathop{\prod }\limits_{{i \in I}}{\operatorname{Hom}}_{R}\left( {M,{A}_{i}}\right) \]\n\nwhich is natural in \( M \) and in \( {\left( {A}_{i}\right) }_{i \in I} \) . | Proof. The projections \( {\pi }_{i} : \mathop{\prod }\limits_{{i \in I}}{A}_{i} \rightarrow {A}_{i} \) induce homomorphisms \( {\operatorname{Hom}}_{R}\left( {M,{\pi }_{i}}\right) : {\operatorname{Hom}}_{R}\left( {M,\mathop{\prod }\limits_{{i \in I}}{A}_{i}}\right) \rightarrow {\operatorname{Hom}}_{R}\left( {M,{A}_{i}... | Yes |
Lemma 2.7. Let \( A \) be an abelian group and let \( M \) be a left \( R \) -module. If \( \varphi \in {\operatorname{Hom}}_{\mathbb{Z}}\left( {M, A}\right) \), then the mapping \( \xi \) that sends \( x \in M \) to \( \xi \left( x\right) : r \mapsto \) \( \varphi \left( {rx}\right) \) is a module homomorphism of \( M... | Proof. \( {\operatorname{Hom}}_{\mathbb{Z}}\left( {{R}_{R}, A}\right) \) is a left \( R \) -module, in which \( \left( {r\alpha }\right) s = \alpha \left( {sr}\right) \) for all \( r, s \in R \) . If \( x \in M \), then \( \xi \left( x\right) : r \mapsto \varphi \left( {rx}\right) \) is in \( {\operatorname{Hom}}_{\mat... | Yes |
Proposition 2.8. If \( A \) is a divisible abelian group, then \( {\operatorname{Hom}}_{\mathbb{Z}}\left( {{R}_{R}, A}\right) \) is an injective left \( R \) -module. | Proof. Let \( \psi : M \rightarrow N \) be any monomorphism of left \( R \) -modules. Since \( D \) is injective as a \( \mathbb{Z} \) -module, \( {\psi }^{ * } : {\operatorname{Hom}}_{\mathbb{Z}}\left( {N, A}\right) \rightarrow {\operatorname{Hom}}_{\mathbb{Z}}\left( {M, A}\right) \) is surjective, by 2.4. Since the d... | Yes |
Proposition 3.6. Let \( \mathcal{A} = \left( {A,\alpha }\right) \) be a direct system of left \( R \) -modules over a directed preordered set \( I \) . A cone \( \varphi : \mathcal{A} \rightarrow M \) is a limit cone of \( \mathcal{A} \) if and only if (i) \( M = \mathop{\bigcup }\limits_{{i \in I}}\operatorname{Im}{\v... | Proof. The limit cone \( \lambda : \mathcal{A} \rightarrow L \) constructed above, \( {\lambda }_{i}\left( x\right) = \operatorname{cls}\left( {x, i}\right) \), has properties (i) \( L = \mathop{\bigcup }\limits_{{i \in I}}{\lambda }_{i}\left( {A}_{i}\right) \) and (iii) \( {\lambda }_{i}\left( x\right) = {\lambda }_{j... | Yes |
Proposition 3.7. Let \( \mathcal{A} = \left( {A,\alpha }\right) \) and \( \mathcal{B} = \left( {B,\beta }\right) \) be direct systems of left \( R \) -modules over the same directed preordered set \( I \), with limit cones \( \lambda : \mathcal{A} \rightarrow L \) and \( \mu : \mathcal{B} \rightarrow M \) . Every homom... | Proof. In the statement, \( \mu \circ \varphi : \mathcal{A} \rightarrow M \) is a cone and factors uniquely through \( \lambda \) . The last parts of the statement follow from this uniqueness. \( ▱ \) | No |
Proposition 3.8. Let \( \varphi : \mathcal{A} \rightarrow \mathcal{B} \) and \( \psi : \mathcal{B} \rightarrow \mathcal{C} \) be homomorphisms of direct systems of left R-modules over the same directed preordered set I. If \( {A}_{i}\overset{{\varphi }_{i}}{ \rightarrow }{B}_{i}\overset{{\psi }_{i}}{ \rightarrow }{C}_{... | Proof. Let \( \lambda : \mathcal{A} \rightarrow L,\mu : \mathcal{B} \rightarrow M \), and \( v : \mathcal{C} \rightarrow N \) be limit cones; let \( \bar{\varphi } = \underline{\lim }{\varphi }_{i},\bar{\psi } = \underline{\lim }{\psi }_{i} \), so that the diagram below commutes for every \( i \in I \) : ![5e708ed9-3d6... | Yes |
Proposition 4.3. Every inverse system of left R-modules has an inverse limit. | Proof. Let \( \mathcal{A} = \left( {A,\alpha }\right) \) be an inverse system of left \( R \) -modules over a directed preordered set \( I \) . We retrieve an inverse limit of \( \mathcal{A} \) from the direct product \( P = \mathop{\prod }\limits_{{i \in I}}{A}_{i} \) and its projections \( {\pi }_{i} : P \rightarrow ... | Yes |
Proposition 4.4. Let \( \mathcal{A} = \left( {A,\alpha }\right) \) be an inverse system of left \( R \) -modules over a directed preordered set \( I \) . A cone \( \varphi : M \rightarrow \mathcal{A} \) is a limit cone of \( \mathcal{A} \) if and only if (i) \( \mathop{\bigcap }\limits_{{i \in I}}\operatorname{Ker}{\va... | Proof. The limit cone \( \lambda : L \rightarrow \mathcal{A} \) constructed in the proof of 4.3 has properties (i) and (ii): if \( x \in L \) and \( {\lambda }_{i}\left( x\right) = 0 \) for all \( i \), then \( x = 0 \) ; if \( {x}_{i} \in {A}_{i} \) and \( {\alpha }_{ij}\left( {x}_{i}\right) = {x}_{j} \) whenever \( i... | Yes |
Proposition 4.6. Let \( \varphi : \mathcal{A} \rightarrow \mathcal{B} \) and \( \psi : \mathcal{B} \rightarrow \mathcal{C} \) be homomorphism of inverse systems of left \( R \) -modules over the same directed preordered set \( I \) . If \( 0 \rightarrow {A}_{i}\overset{{\varphi }_{i}}{ \rightarrow }{B}_{i}\overset{{\ps... | Proof. Let \( \lambda : \mathcal{A} \rightarrow L,\mu : \mathcal{B} \rightarrow M \), and \( v : \mathcal{C} \rightarrow N \) be limit cones; let \( \bar{\varphi } = \underline{\lim }{\varphi }_{i},\bar{\psi } = \underline{\lim }{\psi }_{i} \), so that the diagram below commutes for every \( i \in I \) :\n\n![5e708ed9-... | Yes |
Proposition 5.2. Let \( R \) be a ring, let \( A \) be a right \( R \) -module, let \( B \) be a left \( R \) -module, and let \( C \) be an abelian group. For a mapping \( \beta : A \times B \rightarrow C \) the following conditions are equivalent:\n\n(1) for all \( a,{a}^{\prime } \in A, b,{b}^{\prime } \in B \), and... | Proof. Addition on \( {\operatorname{Hom}}_{\mathbb{Z}}\left( {B, C}\right) \) is pointwise and that \( r \in R \) acts on \( \varphi \in {\operatorname{Hom}}_{\mathbb{Z}}\left( {B, C}\right) \) by \( \left( {\varphi r}\right) \left( b\right) = \varphi \left( {rb}\right) \) for all \( b \in B \) ; hence (1) states that... | Yes |
Proposition 5.3. For every right \( R \) -module \( A \) and left \( R \) -module \( B, A{ \otimes }_{R}B \) and its tensor map exist, and they are unique up to isomorphism. | Proof. Uniqueness follows from the universal property. Existence is proved by constructing a tensor product. Let \( T = F/K \), where \( F \) is the free abelian group on the set \( A \times B \), and \( K \) is the subgroup of \( F \) generated by all \( (a + \) \( \left. {{a}^{\prime }, b}\right) - \left( {a, b}\righ... | Yes |
Every element of \( A{ \otimes }_{R}B \) is a finite sum \( \mathop{\sum }\limits_{i}\left( {{a}_{i} \otimes {b}_{i}}\right) \), where \( {a}_{i} \in A \) and \( {b}_{i} \in B \) . | Let \( T = F/K \) as above. Every element of \( F \) is a finite linear combination \( \mathop{\sum }\limits_{i}{n}_{i}\left( {{a}_{i},{b}_{i}}\right) \) with integer coefficients. Hence every element of \( T = \pi \left( F\right) \) is a finite linear combination \( \mathop{\sum }\limits_{i}{n}_{i}\left( {{a}_{i} \oti... | Yes |
Proposition 5.5. If \( \varphi : A \rightarrow {A}^{\prime } \) is a homomorphism of right \( R \) -modules, and \( \psi : B \rightarrow {B}^{\prime } \) is a homomorphism of left \( R \) -modules, then there is a unique homomorphism \( \varphi \otimes \psi : A{ \otimes }_{R}B \rightarrow {A}^{\prime }{ \otimes }_{R}{B... | Proof. Let \( \tau : A \times B \rightarrow A{ \otimes }_{R}B \) and \( {\tau }^{\prime } : {A}^{\prime } \times {B}^{\prime } \rightarrow {A}^{\prime }{ \otimes }_{R}{B}^{\prime } \) be the tensor maps. Since \( {\tau }^{\prime } \) is a bihomomorphism, \( {\tau }^{\prime } \circ \left( {\varphi \times \psi }\right) :... | Yes |
Proposition 5.7. Let \( A \) be a right \( R \) -module and let \( B \) be a left \( R \) -module.\n\n(1) if \( A \) is an \( S \) - \( R \) -bimodule, then \( A{ \otimes }_{R}B \) is a left \( S \) -module, in which \( s\left( {a \otimes b}\right) = \) \( {sa} \otimes b \) for all \( a \in A, b \in B \), and \( s \in ... | Proof. We prove (1), (2), and (3), and leave the other parts to eager readers. Let \( A \) be an \( S \) - \( R \) -bimodule. If \( s \in S \), then \( {\alpha }_{s} : a \mapsto {sa} \) is a right \( R \) -module endomorphism of \( A \) . By 5.5, there is a unique endomorphism \( {\bar{\alpha }}_{s} = {\alpha }_{s} \ot... | No |
Proposition 5.10. Let \( M \) be a left \( R \) -module and let \( \rho : R \rightarrow S \) be a homomorphism of rings [with identity].\n\n(1) \( S{ \otimes }_{R}M \) is a left \( S \) -module, and \( \iota : x \mapsto 1 \otimes x \) is a homomorphism of left \( R \) -modules of \( M \) into \( S{ \otimes }_{R}M \) ; | Proof. (1). \( S \) is an \( S \) - \( R \) -bimodule, in which \( s \cdot r = {s\rho }\left( r\right) \) for all \( s \in S \) and \( r \in R \) . By 5.7, \( S{ \otimes }_{R}M \) is a left \( S \) -module. Hence \( S{ \otimes }_{R}M \) is also a left \( R \) -module, in which \( r\left( {s \otimes x}\right) = \rho \le... | Yes |
Proposition 6.1. For every right \( R \) -module \( A \) and left \( R \) -module \( B \) there is a commutativity isomorphism \( B{ \otimes }_{{R}^{\text{op }}}A \cong A{ \otimes }_{R}B \), which sends \( b \otimes a \) to \( a \otimes b \) and is natural in \( A \) and \( B \) . If \( A \) and \( B \) are bimodules, ... | Proof. If \( C \) is an abelian group, then \( \beta \) is a bihomomorphism of \( {A}_{R} \times {}_{R}B \) into \( C \) if and only if \( {\beta }^{\mathrm{{op}}} : \left( {b, a}\right) \mapsto \beta \left( {a, b}\right) \) is a bihomomorphism of \( {B}_{{R}^{\mathrm{{op}}}} \times {}_{{R}^{\mathrm{{op}}}}A \) into \(... | No |
For every right R-module \( A, R \) -S-bimodule \( B \), and left \( S \) -module \( C \), there is an associativity isomorphism \( \left( {A{ \otimes }_{R}B}\right) { \otimes }_{S}C \cong A{ \otimes }_{R}\left( {B{ \otimes }_{S}C}\right) \) , which sends \( \left( {a \otimes b}\right) \otimes c \) to \( a \otimes \lef... | We prove the bimodule case and let \( A \) be a \( Q - R \) -bimodule and \( C \) be an \( S \) - \( T \) -bimodule, so that \( \left( {A{ \otimes }_{R}B}\right) { \otimes }_{S}C \) and \( A{ \otimes }_{R}\left( {B{ \otimes }_{S}C}\right) \) are \( Q \) - \( T \) -bimodules; the first part of the statement is the case ... | Yes |
Proposition 6.5. For every \( Q \) -R-bimodule \( A, R \) -S-bimodule \( B \), and \( S \) - \( T \) -bimodule \( C \), a tensor product \( A{ \otimes }_{R}B{ \otimes }_{S}C \) and its tensor map exist, and they are unique up to isomorphism. | To Proposition 6.2 can be added natural isomorphisms \( \left( {A{ \otimes }_{R}B}\right) { \otimes }_{S}C \cong \) \( A{ \otimes }_{R}B{ \otimes }_{S}C \cong A{ \otimes }_{R}\left( {B{ \otimes }_{S}C}\right) \), which send \( \left( {a \otimes b}\right) \otimes c \) to \( a \otimes b \otimes c \) and to \( a \otimes \... | No |
Proposition 6.6 (Adjoint Associativity). Let \( A \) be a right \( R \) -module, let \( B \) be a left \( R \) -module, and let \( C \) be an abelian group. There are adjoint associativity isomorphisms\n\n\[ \Theta : {\operatorname{Hom}}_{\mathbb{Z}}\left( {A{ \otimes }_{R}B, C}\right) \cong {\operatorname{Hom}}_{R}\le... | Proof. The set Bihom \( \left( {A \times B, C}\right) \) of all bihomomorphisms of \( A \times B \) into \( C \) is an abelian group under pointwise addition. The universal property of the tensor map \( \tau : \left( {a, b}\right) \mapsto a \otimes b \) provides a bijection \( \varphi \mapsto \varphi \circ \tau \) of \... | Yes |
Proposition 6.7 (Right Exactness). For every right R-module \( M \) and left \( R \)-module \( N \), the functors \( M{ \otimes }_{R} - \) and \( - { \otimes }_{R}N \) are right exact: if \( A \rightarrow B \rightarrow C \rightarrow 0 \) is exact, then so is\n\n\[ M{ \otimes }_{R}A \rightarrow M{ \otimes }_{R}B \righta... | Proof. We prove the first half of the statement; then 6.1 yields the second half. Let \( A\overset{\varphi }{ \rightarrow }B\overset{\psi }{ \rightarrow }C \rightarrow 0 \) be exact; let \( \bar{\varphi } = {1}_{M} \otimes \varphi \) and \( \bar{\psi } = {1}_{M} \otimes \psi \).\n\n![5e708ed9-3d6d-4f59-a748-eaac13dfd78... | Yes |
Proposition 6.9. There are natural isomorphisms \( M{ \otimes }_{R}\left( {{\bigoplus }_{i \in I}{B}_{i}}\right) \cong \) \( {\bigoplus }_{i \in I}\left( {M{ \otimes }_{R}{B}_{i}}\right) \) and \( \left( {{\bigoplus }_{i \in I}{A}_{i}}\right) { \otimes }_{R}N \cong {\bigoplus }_{i \in I}\left( {{A}_{i}{ \otimes }_{R}N}... | Proof. Let \( M \) be a right \( R \) -module and let \( {\left( {A}_{i}\right) }_{i \in I} \) be a family of left \( R \) -modules. The injection \( {\iota }_{i} : {A}_{i} \rightarrow {\bigoplus }_{i \in I}{A}_{i} \) induces a homomorphism \( {\bar{\iota }}_{i} = {1}_{M} \otimes {\iota }_{i} : M{ \otimes }_{R}{A}_{i} ... | Yes |
Proposition 7.1. If \( F \) is a free left (or right) \( R \) -module with a finite basis \( {\left( {e}_{i}\right) }_{i \in I} \) , then \( {F}^{ * } \) is free with a finite basis \( {\left( {e}_{i}^{ * }\right) }_{i \in I} \), such that \( {e}_{i}^{ * }\left( {e}_{i}\right) = 1,{e}_{i}^{ * }\left( {e}_{j}\right) = 0... | Then \( {\left( {e}_{i}^{ * }\right) }_{i \in I} \) is the dual basis of the given basis \( {\left( {e}_{i}\right) }_{i \in I} \) . Proposition 7.1 does not extend to all free modules: for instance, \( {\left( {\bigoplus }_{i \in I}\mathbb{Z}\right) }^{ * } \cong \mathop{\prod }\limits_{{i \in I}}\mathbb{Z} \) is not f... | No |
Proposition 7.4. If \( M \) is a finitely generated projective module, then the evaluation homomorphism \( M \rightarrow {M}^{* * } \) is an isomorphism. | Proof. If \( F \) is free, with a basis \( {\left( {e}_{i}\right) }_{i \in I} \), then applying 7.1 twice yields a basis \( {\left( {e}_{i}^{* * }\right) }_{i \in I} \) of \( {F}^{* * } \) such that \( {e}_{i}^{* * }\left( {e}_{i}^{ * }\right) = 1,{e}_{i}^{* * }\left( {e}_{j}^{ * }\right) = 0 \) for all \( j \neq i \) ... | Yes |
Proposition 7.5. Let \( A \) and \( B \) be left \( R \) -modules. There is a homomorphism \( \zeta : {A}^{ * }{ \otimes }_{R}B \rightarrow {\operatorname{Hom}}_{R}\left( {A, B}\right) \), which is natural in \( A \) and \( B \), such that \( \zeta \left( {\alpha \otimes b}\right) \left( a\right) = \alpha \left( a\righ... | Proof. For every \( \alpha \in {A}^{ * } \) and \( b \in B,\beta \left( {\alpha, b}\right) : a \mapsto \alpha \left( a\right) b \) is a module homomorphism of \( A \) into \( B \) . We see that \( \beta : {A}^{ * } \times B \rightarrow {\operatorname{Hom}}_{R}\left( {A, B}\right) \) is a bihomomorphism. Hence \( \beta ... | No |
Corollary 7.6. Let \( A \) be a finitely generated projective right \( R \) -module. There is an isomorphism \( A{ \otimes }_{R}B \cong {\operatorname{Hom}}_{R}\left( {{A}^{ * }, B}\right) \), which is natural in \( A \) and \( B \) . | Proof. By 7.4,7.5, \( A{ \otimes }_{R}B \cong {A}^{* * }{ \otimes }_{R}B \cong {\operatorname{Hom}}_{R}\left( {{A}^{ * }, B}\right) \). | No |
Corollary 7.7. Let \( R \) be commutative and let \( A, B \) be finitely generated projective \( R \) -modules. There is an isomorphism \( {A}^{ * }{ \otimes }_{R}{B}^{ * } \cong {\left( A{ \otimes }_{R}B\right) }^{ * } \), which is natural in \( A \) and \( B \) . | Proof. By 7.5,6.6, \( {A}^{ * }{ \otimes }_{R}{B}^{ * } \cong {\operatorname{Hom}}_{R}\left( {A,{B}^{ * }}\right) = {\operatorname{Hom}}_{R}\left( {A,{\operatorname{Hom}}_{R}\left( {B, R}\right) }\right) \) \( \cong {\operatorname{Hom}}_{R}\left( {A{ \otimes }_{R}B, R}\right) = {\left( A{ \otimes }_{R}B\right) }^{ * }.... | Yes |
Proposition 8.1. Every projective module is flat. | Proof. Readers will verify that free modules are flat. Now, let \( P \) be a projective left \( R \) -module. There exist a free left \( R \) -module \( F \) and homomorphisms \( \pi : F \rightarrow \) \( P,\iota : P \rightarrow F \) such that \( \pi \circ \iota = {1}_{P} \) . Every monomorphism \( \mu : A \rightarrow ... | No |
Proposition 8.6. An abelian group is flat (as a \( \mathbb{Z} \) -module) if and only if it is torsion-free. | Proof. Finitely generated torsion-free abelian groups are free, and are flat by 8.1. Now every torsion-free abelian group is the direct limit of its finitely generated subgroups, which are also torsion-free, and is flat by 8.4.\n\nOn the other hand, finite cyclic groups are not flat: if \( C \) is cyclic of order \( m ... | Yes |
Proposition 8.8. Every module is a direct limit of finitely presented modules. | Proof. Let \( M \cong F/K \), where \( F \) is free with a basis \( X \) . We show that \( M \) is the direct limit of the finitely presented modules \( {F}_{Y}/S \), where \( {F}_{Y} \) is the free submodule of \( F \) generated by a finite subset \( Y \) of \( X \) and \( S \) is a finitely generated submodule of \( ... | Yes |
Corollary 8.10. Every finitely presented flat module is projective. | Proof. The identity on such a module factors through a free module. | No |
Theorem 8.11 (Lazard [1969]). For a left R-module \( M \) the following conditions are equivalent:\n\n(1) \( M \) is flat;\n\n(2) every homomorphism of a finitely presented free module into \( M \) factors through a finitely generated free module;\n\n(3) \( M \) is a direct limit of finitely generated free modules;\n\n... | Proof. (3) implies (4); (4) implies (1), by 8.1 and 8.4; (1) implies (2), by 8.9. Now assume (2). Let \( \pi : F \rightarrow M \) be an epimorphism, where \( F \) is free with a basis \( X \) . Choose \( \pi : F \rightarrow M \) so that there are for every \( m \in M \) infinitely many \( x \in X \) such that \( \pi \l... | Yes |
Lemma 9.1. Let \( \mathfrak{a} \) be an ideal of a commutative ring \( R \), let \( M \) be an \( R \) -module, and let \( {M}_{1} \supseteq {M}_{2} \supseteq \cdots \) be an \( \mathfrak{a} \) -filtration on \( M \). (1) \( {R}^{ + } = R \oplus \mathfrak{a} \oplus {\mathfrak{a}}^{2} \oplus \cdots \) is a ring (the blo... | Proof. (1). The elements of \( {R}^{ + } \) are infinite sequences \( a = \left( {{a}_{0},{a}_{1},\ldots ,{a}_{i}}\right. \) , \( \ldots ) \) in which \( {a}_{0} \in R,{a}_{i} \in {\mathfrak{a}}^{i} \) for all \( i > 0 \), and \( {a}_{i} = 0 \) for almost all \( i \) . Addition in \( {R}^{ + } \) is componentwise; mult... | Yes |
Lemma 9.2 (Artin-Rees). Let \( \mathfrak{a} \) be an ideal of a commutative Noetherian ring \( R \), and let \( M \) be a finitely generated \( R \) -module. If \( {M}_{1} \supseteq {M}_{2} \supseteq \cdots \) is an \( \mathfrak{a} \) -stable filtration on \( M \), then \( N \cap {M}_{1} \supseteq N \cap {M}_{2} \supse... | Proof. First, \( N \cap {M}_{1} \supseteq N \cap {M}_{2} \supseteq \cdots \) is an \( \mathfrak{a} \) -filtration on \( N \) ; \( M \) is a Noetherian \( R \) -module, by VIII.8.5; \( \bar{N} \) and all \( {M}_{i}, N \cap {M}_{i} \) are finitely generated; and \( {N}^{ + } \) is an \( {R}^{ + } \) -submodule of \( {M}^... | Yes |
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