By Catarina Santa-Clara, Alberto Facchini, Kent R. Fuller, Claus Michael Ringel

Surveying the main influential advancements within the box, this lawsuits reports the newest learn on algebras and their representations, commutative and non-commutative earrings, modules, conformal algebras, and torsion theories. the amount collects stimulating discussions from world-renowned names together with Tsit-Yuen Lam, Larry Levy, Barbara Osofsky, and Patrick Smith.

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A module M is said to be A-c-injective if for every closed submodule B of A, every homomorphism B —> M extends to a homomorphism A —> M. A module M is called self-c-injective if it is M-c-injective. 6. We are going to see that r-torsion modules are even r-divisible under some extra hypotheses. 8. Let r be stable. Then every r-torsion c-injective module is r-divisible. Proof. Let D be a r-torsion c-injective module. Also, let A be a module, B be a r-closed submodule of A and u: B —» D be a non-zero monomorphism.

We conclude (1)=0, hence either [I,T] ® / = 0 or [J,T] © J = 0 by gr-primeness of L. Therefore T is prime. 2. We just need to repeat the proof of 1 for J = I. 3. We claim Ann T (T) C Ann L (L). (3) Indeed, let x G Ann T (T). 7 yields [x, L\] = 0. Since we also have for any y G LQ, (y = ^2[zi,U], Zi,U G T), that [x, y] = — J2izi> h, x] — 0 we conclude (3). If we now take x = XQ + x\ G L such that x G Ann/,(L), we have by gradedness that [xi,T] = [xo,T] = 0. 7, XQ = 0. Hence Ann L (L) c Ann T (T).

Let Ai and Ai be modules such that A\ 0 Ai is self-rdivisible. , are both self-r-divisible and relatively r-divisible. Proof. 5. 6. Let A\ and Ai be modules. 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.

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