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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 Fall '08
 EVERAGE
 Algebra, Vector Space, Ring, unital KŒx–module

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