Module Theory Claudia Menini 2014-2015 Contents Contents 2 1 Modules 4 1.1 Homomorphisms and Quotients ..................... 4 1.2 Quotient Module and Isomorphism Theorems . 10 1.3 Product and Direct Sum of a Family of Modules . 13 1.4 Sum and Direct Sum of Submodules. Cyclic Modules . 21 1.5 Exact sequences and split exact sequences . 27 1.6 HomR (M; N) ............................... 32 2 Free and projective modules 35 3 Injective Modules and Injective Envelopes 41 4 Generators and Cogenerators 59 5 2 × 2 Matrix Ring 65 6 Tensor Product and bimodules 68 6.1 Tensor Product 1 ............................. 68 6.2 Bimodules ................................. 78 6.3 Tensor Product 2 ............................. 82 7 Homology 94 7.1 Categories and Functors ......................... 94 7.2 Snake Lemma ...............................100 7.3 Chain Complexes .............................105 7.4 Homotopies ................................113 7.5 Projective resolutions . 116 7.6 Left Derived functors . 123 7.7 Cochain Complexes and Right Derived Functors . 139 8 Semisimple modules and Jacobson radical 145 9 Chain Conditions. 160 2 CONTENTS 3 10 Progenerators and Morita Equivalence 173 10.1 Progenerators ...............................173 10.2 Frobenius .................................199 Chapter 1 Modules 1.1 Homomorphisms and Quotients R Definition 1.1. Let R be a ring. A left R-module is a pair (M; µM ) where (M+; 0) is an abelian group and R µ = µM : R × M ! M is a map such that, setting a · x = µ ((a; x)) , the following properties are satisfied : M1 a · (x + y) = a · x + a · y; M2 (a + b) · x = a · x + b · x; M3 (a ·R b) x = a · (b · x); M4 1R · x = x for every a; b 2 R and every x; y 2 M. In this case we will say that M is a left R-module. The notation RM will be used to mean that M is a left R-module. Definition 1.2. Let R be a ring and let Rop denote the opposite ring of R.A right R-module is a left Rop-module i.e. it is a pair (M; µ0) where (M+; 0) is an abelian group and 0 R × ! µ = µM : R M M is a map such that, setting a · x = µ0 ((a; x)) , M10 a · (x + y) = a · x + a · y; 4 1.1. HOMOMORPHISMS AND QUOTIENTS 5 M20 (a + b) · x = a · x + b · x; 0 M3 (a ·Rop b) · x = a · (b · x); 0 M4 1R · x = x for every a; b 2 R and every x; y 2 M. In this case we will say that M is a right R-module. The notation MR will be used to mean that M is a right R-module. Note that a ·Rop b = b ·R a so that M3' rewrites as (a ·R b) · x = (b ·Rop a) · x = b · (a · x) For this reason, if M is a right R-module, one usually writes x · a instead of a · x, for every a 2 R, x 2 M. With this notation the conditions M1'), M2'), M3'), M4') may be rephrased as follows: M1" (x + y) · a = x · a + y · a; M2" x · (a + b) = x · a + x · b; M3" x · (a ·R b) = (x · a) · b; M4" x · 1R = x. The abelian group M is called the underlying additive group of the left ( resp. right) R-module M. Given x; y 2 M we will write x − y instead of x + (−y). Remark 1.3. If R is a commutative ring, then every left R-module is, in a natural way, a right R-module, and conversely. In fact, let M be a left R-module, given a; b 2 R, x 2 M, we have a · (b · x) = (a ·R b) x = (b ·R a) x = b · (a · x) . In the same way, if M is a right R-module, given a; b 2 R, x 2 M, we have: (x · a) · b = x (a ·R b) = x · (b ·R a) = (x · b) · a. Therefore, when R is a commutative ring, we will, in general, simply say that M is an R-module. Examples 1.4. 6 CHAPTER 1. MODULES 1. Let G be an abelian group with additive notation. G becomes, in a natural way, a Z- module by defining, for every n 2 Z and x 2 G, n · x = nx where nx denotes the nth power of x in the additive notation. 2. Let A be a ring, R be a subring of A. A becomes a left (resp. right) R-module by setting, for every r 2 R, a 2 A, ra ( resp. ar) to be the product of the element r 2 R ⊆ A with the element a 2 A ( resp. of the element a 2 A with the element r 2 R ⊆ A) in the ring A. In particular the rings R; R[X];R[[X]] may be considered as left (resp. right) R-modules. If D is a commutative domain, Q (D) is a D-module. 3. (More generally,) let f : R ! A be a ring homomorphism. Any left A-module A M; µM inherits the structure of a left R-module by setting R A µM ((r; x)) = µM ((f (r) ; x)) for every r 2 R and x 2 M i.e. r · x = f (r) · x for every r 2 A and x 2 M: This module is often denoted by f∗ (M) and called the R-module obtained by restriction of the ring of scalars from A to R. 1.5. If R is a division ring and M is a left (resp. right) R-module we say that M is a left (resp. right) vector space over R. If R is a field, we simply say that M is a vector space over R. Proposition 1.6. Let R be a ring, M a left R-module. Then, for every a; b 2 R and for every x; y 2 M we have : 1. a · 0M = 0M ; 2. 0R · x = 0M ; 3. (−a) · x = −a · x = a · (−x) ; (−a) · (−x) = a · x; 4. a · (x − y) = a · x − a · y; 5. (a − b) · x = a · x − b · x . 6. n (a · x) = (na) · x = a · (nx) for every n 2 Z; a 2 R; x 2 M: 1.1. HOMOMORPHISMS AND QUOTIENTS 7 Proof. 1) Let us start from : a·0M = a (0M + 0M ) = a·0M +a·0M : Adding − (a · 0M ) to both sides we find : 0M = a · 0M : 2) First we look at the obvious : 0R · x = (0R + 0R) x = 0R · x + 0R · x: Adding − (x · 0R) to both sides we find : 0M = 0R · x: 3) From (−a) x + ax = ((−a) + a)x = 0R · x = 0M we obtain that (−a) x = −ax. In a similar way it follows from ax + a (−x) = a(x + (−x)) = a · 0M = 0M that a (−x) = −ax. Moreover : (−a)(−x) = −(a(−x)) = −(−(ax)) = ax: 4) We calculate: a(x − y) = a(x + (−y)) = ax + a(−y) = ax + (−(ay)) = ax − ay. 5) We calculate: (a − b) x = (a + (−b))x = ax + (−b) x = ax + (−bx) = ax − bx. 6) It is easily proved by Induction. 1.7. Let M be an abelian group and let A = End(M) denote the ring of endomor- phisms of M. Then M becomes a left A-module by setting f · x = f (x) every f 2 A and x 2 M: In fact,note that (f ·A g) x = (f ◦ g)·x = (f ◦ g)(x) = f (g (x)) = f·(g · x) for every f; g 2 A and x 2 M: Now let ' : R ! End(M) be a ring morphism. Then, in view of Example 3 in 1.4, we can consider the left R-module '∗ (M) i.e. M becomes a left R-module by setting r · m = '(r)(m) for all r 2 R and for all m 2 M. Conversely let M be a left R-module and let End(M) denote the ring of endomor- phisms of the abelian group underlying the R-module structure of M. For every r 2 R consider the map t : M ! M r . m 7! r · m Clearly tr 2 End(M) and the map : R ! End(M) r 7! tr is a ring morphism. In this way we get: Theorem 1.8. Let R be a ring and let M be an abelian group. The ring morphisms ' : R ! End(M) correspond bijectively to the left R-module structures on M. 8 CHAPTER 1. MODULES Proof. Using notation as above, given a ring morphisms ' : R ! End(M) we have: (r)(m) = r · m = '(r)(m). Conversely, if M is a left R-module we have: r · m = (r)(m). To get an analogous result for right R-modules we have to consider the ring End(M)op which has the same addition as End(M) but where multiplication is defined by f · g = g ◦ f. Rephrasing the foregoing theorem we obtain : Theorem 1.9. Let R be a ring and let M be an abelian group. The ring morphisms ' : R ! End(M)op correspond bijectively to the right R-module structures on M. Definitions 1.10. Let R be a ring and let M be a left R-module. A subset N of M is said to be an R-submodule ( or simply submodule) of M if : 1. N is a subgroup of M; 2. a 2 R and x 2 N implies that a · x 2 N, for every a 2 R and x 2 N. We write N ≤R M to mean that N is a submodule of M. We denote by L(RM) the set of all the submodules of RM.