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f09_mthsc852_hw17

# f09_mthsc852_hw17 - R-module D the sequence → Hom R N,D...

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MTHSC 851/852 (Abstract Algebra) Dr. Matthew Macauley HW 17 Due Monday, November 16th, 2009 (1) If R is a ring with 1 and M is an R -module that is not unitary, show that Rm = 0 for some m 6 = 0. (2) If F is a ﬁeld set R = F [ x 1 ,x 2 ,x 3 ,... ], the ring of polynomials in a countably inﬁnite set of distinct indeterminantes. Let I be the ideal ( x 1 ,x 2 ,... ) in R . If M = R and N = I show that M is a ﬁnitely generated R -module but N is a submodule that is not ﬁnitely generated. Is N free? (3) Suppose L , M and N are R -modules and f : M N is an R -homomorphism. Deﬁne f * : Hom R ( N,L ) Hom R ( M,L ) via f * ( φ ) : m 7→ φ ( f ( m )) for all φ Hom R ( N,L ), m M . (a) Show that f * is a Z -homomorphism. (b) If R is commutative show that f * is an R -homomorphism. (c) Still assuming that R is commutative, show that if 0 L f -→ M g -→ N 0 is an exact sequence of R -modules, then for any
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Unformatted text preview: R-module D , the sequence → Hom R ( N,D ) g *-→ Hom R ( M,D ) f *-→ Hom R ( L,D ) is an exact sequence of abelian groups. (4) The Five Lemma states that given a diagram of abelian groups A 1 / f 1 ± A 2 / f 2 ± A 3 / f 3 ± A 4 / f 4 ± A 5 f 5 ± A 1 / A 2 / A 3 / A 4 / A 5 where the rows are exact, and f 1 ,f 2 ,f 4 and f 5 are isomorphisms, f 3 is an isomorphism as well. (a) Prove the Five Lemma. (b) Consider the following eight hypotheses: f i is injective, for i = 1 , 2 , 4 , 5 , f i is surjective, for i = 1 , 2 , 4 , 5 . Which of these hypothese suﬃce to prove that f 3 is injective? Which suﬃce to prove that f 3 is surjective?...
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