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Unformatted text preview: MATH 8020 CHAPTER 3: MODULES PETE L. CLARK Contents 3. The category of modules over a ring 1 3.1. Basic definitions 1 3.2. Finitely presented modules 6 3.3. Torsion and torsionfree modules 8 3.4. Tensor products of modules 9 3.5. Projective modules 11 3.6. Injective modules 18 3.7. Tor functors 26 3.8. Flat modules 26 3.9. Nakayama’s Lemma 27 3.10. Ordinal Filtrations and Applications 29 3.11. More on ﬂat modules 33 References 37 3. The category of modules over a ring 3.1. Basic definitions. Suppose ( M, +) is an abelian group. For any m ∈ M and any integer n , one can make sense of n • m . If n is a positive integer, this means m + ··· + m ( n times); if n = 0 it means 0, and if n is negative, then n • m = − ( − n ) • m . Thus we have defined a function • : Z × M → M which enjoys the following properties: for all n, n 1 , n 2 ∈ Z , m, m 1 , m 2 ∈ M , we have (ZMOD1) 1 • m = m . (ZMOD2) n • ( m 1 + m 2 ) = n • m 1 + n • m 2 . (ZMOD3) ( n 1 + n 2 ) • m = n 1 • m + n 2 • m. (ZMOD4) ( n 1 n 2 ) • m = n 1 • ( n 2 • m ) It should be clear that this is some kind of ringtheoretic analogue of a group action on a set. In fact, consider the slightly more general construction of a monoid ( M, · ) acting on a set S : that is, for all n 1 , n 2 ∈ M and s ∈ S , we require 1 • s = s and ( n 1 n 2 ) • s = n 1 • ( n 2 • s ). For a group action G on S , each function g • : S → S is a bijection. For monoidal actions, this need not hold for all elements: e.g. taking the natural multiplication 1 2 PETE L. CLARK action of M = ( Z , · ) on S = Z , we find that 0 • : Z → { } is neither injective nor surjective, ± 1 • : Z → Z is bijective, and for  n  > 1, n • : Z → Z is injective but not surjective. Exercise 3.1: Let • : M × S → S be a monoidal action on a set. Show that for each unit m ∈ M – i.e., an element for which there exists m ′ with mm ′ = m ′ m = 1 – m • : S → S is a bijection. Then the above “action” of Z on an abelian group M is in particular a monoidal action of ( Z , · ) on the set M . But it is more: M has an additive structure, and (ZMOD2) asserts that for each n ∈ Z , n • respects this structure – i.e., is a ho momorphism of groups; also (ZMOD3) is a compatibility between the additive structure on Z and the additive structure on M . These axioms can be restated in a much more compact form. For an abelian group M , an endomorphism of M is just a group homomorphism from M to itself: f : M → M . We write End( M ) for the set of all endomorphisms of M . But End( M ) has lots of additional structure: for f,g ∈ End( M ) we define f + g ∈ End( M ) by ( f + g )( m ) := f ( m ) + g ( m ) , i.e., pointwise addition. We can also define f · g ∈ End( M ) by composition: ( f · g )( m ) := f ( g ( m )) ....
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 Summer '11
 Clark
 Math, Algebra, Ring, pete l. clark, Rmodule

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