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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.
- Spring '11