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College Algebra Exam Review 335

# College Algebra Exam Review 335 - r 2 R and m 2 M Then the...

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Chapter 8 Modules 8.1. The idea of a module Recall that an action of a group G on a set X is a homomorphism ' W G ! Sym .X/: Equivalently, one can view an action as a “product” G X ! X , de- fined in terms of ' by gx D '.g/.x/ , for g 2 G and x 2 X . The homomorphism property of ' translates into the mixed associative law for this product: .g 1 g 2 /x D g 1 .g 2 x/; for g 1 ; g 2 2 G and x 2 X . There is an analogous notion of an action of a ring R on an abelian group M . Definition 8.1.1. An action of a ring R on an abelian group M is a homo- morphism of ' W R ! End .M/ . Given an action ' of R on M , we can define a “product” R M ! M in terms of ' by rm D '.r/.m/ for r
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Unformatted text preview: r 2 R and m 2 M . Then the homo-morphism property of ' translates into mixed associative and distributive laws: .r 1 r 2 /m D r 1 .r 2 m/ and .r 1 C r 2 /m D r 1 m C r 2 m: Moreover, '.r/ 2 End .M/ translates into the second distributive law: r.m 1 C m 2 / D rm 1 C rm 2 : Conversely, given a product R ² M ±! M satisfying the mixed as-sociative law and the two distributive laws, for each r 2 R , deﬁne the map '.r/ W M ! M by '.r/.m/ D rm . Then the second distributive law says 345...
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