8.2. HOMOMORPHISMS AND QUOTIENT MODULES 353 8.1.8. Let V be an n –dimensional vector space over a ﬁeld K . Show that V n (the direct sum of n copies of V ) is a free module over End K .V / . 8.1.9. Let V be a ﬁnite dimensional vector space V over a ﬁeld K . Let T 2 End K .V / . Give V the corresponding KŒxŁ –module structure deﬁned by P i ˛ i x i v D P i ˛ i T i .v/ . Show that V is not free as a KŒxŁ –module. 8.1.10. Show that conditions (a) and (c) in Proposition 8.1.28 are equiva-lent. 8.2. Homomorphisms and quotient modules In this section, we construct quotient modules and develop homomor-phism theorems for modules, which are analogues of the homomorphism theorems for groups and rings. Recall that if M and N are R –modules, then Hom .M;N/ denotes the set of R –module homomorphisms from M to N . Proposition 8.2.1. (a) If ' 2 Hom R .M;N/
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