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College Algebra Exam Review 337

College Algebra Exam Review 337 - V is in particular a...

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8.1. THE IDEA OF A MODULE 347 Example 8.1.6. Any left ideal M in a ring R is a module over R (with the product R ± M ²! M being the product in the ring.) In particular, R is a module over itself. Example 8.1.7. For any ring R , and any natural number n , the set R n of n –tuples of elements of R is an R –module with component-by-component addition and multiplication by elements of R . Example 8.1.8. Any abelian group A is a unital Z –module, with the prod- uct Z ± A ²! A given by .n;a/ 7! na D the n th power of a in the abelian group A . Example 8.1.9. A vector space V over a field K is a module over the ring End K .V / , with the module action given by T v D T.v/ for T 2 End K .V / and v 2 V . Example 8.1.10. Let T be a linear map defined on a vector space V over a field K . Recall from Example 6.2.9 that there is a unital homomorphism from KŒxŁ to End K .V / ' T . X i ˛ i x i / D X i ˛ i T i : This homomorphism makes V into a unital KŒxŁ –module. Conversely, suppose V is a unital KŒxŁ –module, and let ' W KŒxŁ ²! End .V / be the corresponding homomorphism. Then,
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Unformatted text preview: V is, in particular, a unital K –module, thus a K –vector space. For ˛ 2 K and v 2 V , we have ˛v D '.˛/.v/ . Set T D '.x/ 2 End .V / . We have T.˛v/ D '.x/'.˛/.v/ D '.˛/'.x/.v/ D ˛.T v/ for all ˛ 2 K and v 2 V . Thus T is actually a linear map. Moreover, we have '. X i ˛ i x i /v D X i ˛ i T i .v/; so the given unital KŒxŁ –module structure on V is the same as the unital KŒxŁ –module structure arising from the linear map T . What we have called an R –module is also known as a left R –module. One can define a right R –module similarly. Definition 8.1.11. A right module M over a ring R is an abelian group M together with a product M ± R ²! M satisfying m.r 1 r 2 / D .mr 1 /r 2 ; m.r 1 C r 2 / D mr 1 C mr 2 ; and .m 1 C m 2 /r D m 1 r C m 2 r:...
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