# Algebra2009Jan - R-module is called divisible if given a...

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Qualifying Exam - January 2009 Algebra Part Instructions: Complete as many questions as possible. Answers should be justified with the necessary proofs. All rings are assumed to be noncommutative unless stated otherwise. All rings have an identity element and all modules are unitary. 1. (a) Let 1 be a finitely generated left ideal of the ring R and assume S = {rna I a E: X} C I generates 1 as a left ideal. Show that S contains a finite subset that generates 1 as a left ideal of R. (b) Give an example of a ring with a left ideal that can be generated by one element but also has a minimal set of generators containing two elements. 2. Recall that an ideal ofa ring R is called left primitive ifit is the annihilator ofa simple left R-module. (a) Show that a left primitive ideal must be prime. (b) Give an example of a prime ideal that is not left primitive. 3. An element of a ring R is called left regular if its left annihilator is zero and a left
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Unformatted text preview: R-module is called divisible if given a left regular element a E Rand mE M, we can find n E: M such that an = m. Show that any injective left R-module is divisible. 4. Let Z be the ring of integers. For any Z-module A, let F(A) = Homz(A,Z) and for any Z-homomorphism <p: A --B let cp*: F(B) --F(A) be given by cp * (f) = f cp. Consider the short exact sequence 0 -- Z ~ Z---!!.- Z2 --0 where a( n) = 2n for n* a* all n in Z. Is the sequence 0 --F(Zz) ~F(Z) ~ F(Z) --0 exact? Justify your answer. 5. Let Rand S be rings, assume P is a projective left S-module and M is an R-S-bimodule that is projective as a left R-module. Show that M @s P is projective as a left R-module. 6. (a) Suppose J + 1 = R, where I is a left ideal ofRand J = J(R) is the Jacobson radical of R. Show that 1= R. (b) Let L be a left ideal of R such that L + 1= R for any left ideall of R implies 1 = R. Show that L ~ J(R)....
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## This note was uploaded on 06/19/2011 for the course MATH 601 taught by Professor Na during the Spring '11 term at University of North Carolina School of the Arts.

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