# sf - θ to an R-map φ P → M speciﬁcally αφ = θβ...

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Math 5125 Wednesday, November 30 Sample Final Problems since Second Test 1. Let F be a ﬁeld with characteristic 6 = 2. (a) Prove that F [ x ] / ( x - 1 ) = F [ x ] / ( x + 1 ) as rings. (b) Prove that F [ x ] / ( x - 1 ) ± F [ x ] / ( x + 1 ) as F [ x ] -modules. (c) Is F [ x ] / ( x - 1 ) = F [ x ] / ( x + 1 ) as F [ x 2 ] -modules? 2. Let R be a ring with a 1, let M be a left R -module, let I , J be ideals of R , and N = M / IM . Prove that N / JN = M / ( IM + JM ) and JN = JM / ( IM JM ) . 3. Let R be a ring with a 1, let M be a left R -module, and let I be an ideal of R . Prove that R / I R M = M / IM as left R -modules. 4. Let S be a subring of the ring R . Prove or give a counterexample to the following statements. (a) If P is a projective S -module, then R S P is a projective R -module. (b) If P is an injective S -module, then R S P is an injective R -module. 5. Let R be a ring, let J be an ideal of R , and let M , P be R -modules with P projective. Let α : M M / JM and β : P P / JP denote the natural epimorphisms. Suppose θ : P / JP M / JM is an R -map. Prove that we may lift
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Unformatted text preview: θ to an R-map φ : P → M ; speciﬁcally αφ = θβ . Also show by example that the hypothesis P projective cannot be dropped. 6. Let R be a ring with a 1 and let e be an idempotent of R (that means e 2 = e ). Prove that Re is a projective R-module. 7. Let R be a ring with a 1 and let Q be an injective left R-module. Suppose x ∈ R is not a right zero divisor, that is rx 6 = 0 whenever 0 6 = r ∈ R . Prove that xQ = Q . Exam on Tuesday December 13, 10:05 a.m.–12:05 p.m. in the regular classroom McBryde 202. One of the problems will be identical to one of the ungraded homework problems or sample ﬁnal problems since the second test....
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