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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 Kx module structure on V is the same as the unital Kx 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 dene a right R module similarly. Denition 8.1.11. A right module M over a ring R is an abelian group M together with a product M R ! 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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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
- Fall '08