Algebras, Rings And Their Representations: Proceedings Of by Catarina Santa-Clara PDF

By Catarina Santa-Clara

ISBN-10: 9812565981

ISBN-13: 9789812565983

ISBN-10: 9812774556

ISBN-13: 9789812774552

Surveying the main influential advancements within the box, this complaints experiences the most recent examine on algebras and their representations, commutative and non-commutative jewelry, modules, conformal algebras, and torsion theories. the quantity collects stimulating discussions from world-renowned names together with Tsit-Yuen Lam, Larry Levy, Barbara Osofsky, and Patrick Smith.

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Additional resources for Algebras, Rings And Their Representations: Proceedings Of The International Conference on Algebras, Modules and Rings, Lisbon, Portugal, 14-18 July 2003

Example text

In the second case we similarly obtain J = 0. Hence L is gr-prime. Conversely, let us now suppose L is gr-prime and let I, J be two ideals of T satisfying [I, T, J] + [J, T, I] = 0. It is easy to check that [I, T]®I and [J,T}®J are gr-ideals of L. We assert [[/,T]@I, [J,T)®J} = 0. Indeed, for any x S [/,T] ® I we have x = ( X S i £(xi'x'i)>z) w ^ n z>Xi £ I, x't £T for i = 1 , . . , ny. ly3>ypx'i\>xi)-c([yj>y'j>xi}>x'i))- E (2) »ii=l We have, for any u e T , i e { 1 , . . , n x } and j G { 1 , .

If a module is A\ -r-divisible and Ai-injective, then it is (A\ © A2)-r-divisible. Proof. Denote A = A\ © A2 and let D be an Ai-r-divisible and y^-injective module. Let B b e a r-closed submodule of A and u: B —> D be a homomorphism. Let / be the restriction of u to BC\A\ ,i:B^>A and j : B<~\A\ —> A\ be the inclusions and i\: A\ —> A be the canonical injection. Then BC\Ai is r-closed in Ai, because A\/{B n A{) = (B + A{)/B C A/B. Since D is Ai-r-divisible, there exists a homomorphism v. A\ —* D such that vj = / .

Also, we get B C C, because for every b G B we have b = pi(b) + p2{b) = Pif~1P2(b) +P2{b) = vp2(b) +P2(b) G C. (ii) =$» (i) Suppose now that (ii) holds. Let M be a r-closed submodule of A2 and let u: M —> Ai be a homomorphism. Put £? = {m—u(m) | m G M } . It is easy to check that B is a submodule of A and BC\Ai = 0. Also, we have P2(B) = M, hence P2(B) is a r-closed submodule of A2. Now there exists a submodule C of A such that A = A\ ©C and B C C. Let p : A —» Ai denote the projection with kernel C and let q: A2 —> J4I be the restriction of p to J42- Then for every m G M, g(m) = p(m) = p(m — u(m)) + u(m) = u(m).

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Algebras, Rings And Their Representations: Proceedings Of The International Conference on Algebras, Modules and Rings, Lisbon, Portugal, 14-18 July 2003 by Catarina Santa-Clara


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