Unformatted text preview: R –module endomorphism 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 number n , consider R n as the set of n –by– 1 matrices over R (column “vectors”). Let T be a ﬁxed n –by– m matrix over R . Then left multiplication by T is an Rmodule 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 Rmodule homomorphism even if R is noncommutative....
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 Fall '08
 EVERAGE
 Algebra, Vector Space, Ring, Homomorphism, hS iC RS

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