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College Algebra Exam Review 339

College Algebra Exam Review 339 - R –module endomor-phism...

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8.1. THE IDEA OF A MODULE 349 (b) Let h S i be the subgroup of M generated by S . Then h S i C R S is a submodule of M containing S . (c) h S i C R S is the smallest submodule of M containing S . (d) If R has an identity element and M is unital, then S R S , and h S i C R S D R S . The reader is asked to verify these assertions in Exercise 8.1.6 . Definition 8.1.19. R S is called the submodule of M generated by S or the span of S . If x 2 M , then Rx D R f x g is called the cyclic submodule generated by x . If there is a finite set S such that M D R S , we say that M is finitely generated . If there is an x 2 M such that M D Rx , we say that M is cyclic . Remark 8.1.20. Either R S or h S iC R S have a good claim to be called the submodule of M generated by S . Fortunately, in the case in which we are chiefly interested, when R has an identity and M is unital, they coincide. Homomorphisms Definition 8.1.21. Let M and N be modules over a ring R . An R –module homomorphism ' W M ! N is a homomorphism of abelian groups such that '.rm/ D r'.m/ for all r 2 R and m 2 M . An R –module isomor- phism is a bijective R –module homomorphism. An
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Unformatted text preview: R –module endomor-phism of M is an R –module homomorphism from M to M . Notation 8.1.22. The set of all R –module homomorphisms from M to N is denoted by Hom R .M;N/ . The set of all R –module endomoprhisms of M is denoted by End R .M/ . The kernel of an R module homomorphism ' W M ²! N is f x 2 M W '.x/ D g . Example 8.1.23. Suppose R is a commutative ring. For any natural num-ber n , consider R n as the set of n –by– 1 matrices over R (column “vec-tors”). Let T be a fixed n –by– m matrix over R . Then left multiplication by T is an R-module homomorphism from R m to R n . Example 8.1.24. Fix a ring R . Let T be a fixed n –by– m matrix with entries in Z . Then left multiplication by T maps R m to R n , and is an R-module homomorphism even if R is non-commutative....
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