College Algebra Exam Review 338

# College Algebra Exam Review 338 - of V which are invariant...

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348 8. MODULES Example 8.1.12. A right ideal M in a ring R is a right R module. - Example 8.1.13. Let R be the ring of n –by– n matrices over a ﬁeld K . Then, for any s , the vector space M of n –by– s matrices is a left R module, with R acting by matrix multiplication on the left. Similarly, the vector space N of s –by– n matrices is a right R module, with R acting by matrix multiplication on the right. Submodules Deﬁnition 8.1.14. Let R be a ring and let M be an R –module. An R submodule of M is an abelian subgroup W such that for all r 2 R and all w 2 W , rw 2 W . Example 8.1.15. Let R act on itself by left multiplication. The R –submodules of R are precisely the left ideals of R . Example 8.1.16. Let V be a vector space over K and let T 2 End K .V / be a linear map. Give V the structure of a unital KŒxŁ –module as in Example 8.1.10 . Then the KŒxŁ –submodules of V are the linear subspaces W
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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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