Unformatted text preview: of V which are invariant under T ; i.e., T.w/ 2 W for all w 2 W . For example, the kernel and range of T are KŒxŁ –submodules. The reader is asked to verify these assertions in Exercise 8.1.2 . Proposition 8.1.17. Let M be an R –module. (a) Let f M ˛ g be any collection of submodules of M . Then \ ˛ M ˛ is a submodule of M . (b) Let M n be an increasing sequence of submodules of M . Then [ n M n is a submodule of M . (c) Let A and B be two submodules of M . Then A C B D f a C b W a 2 A and b 2 B g is a submodule of M . Proof. Exercise 8.1.5 . n Example 8.1.18. Let M be an R –module and S ± M . (a) Deﬁne R S D f r 1 s 1 C ²²² C r n s n W n 2 N ;r i 2 R;s i 2 S g : Then R S is a submodule of M ....
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 Fall '08
 EVERAGE
 Algebra, Matrices, Vector Space, Ring

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