On perfect simple-injective rings

Harada calls a ring right simple-injective if every -homomorphism with simple image from a right ideal of to is given by left multiplication by an element of . In this paper we show that every left perfect, left and right simple-injective ring is quasi-Frobenius, extending a well known result of Osofsky on self-injective rings. It is also shown that if is left perfect and right simple-injective, then is quasi-Frobenius if and only if the second socle of is countably generated as a left -module, extending many recent results on self-injective rings. Examples are given to show that our results are non-trivial extensions of those on self-injective rings.

     

On graded Thierrin radicals of graded rings

In this paper, we study the graded Thierrin radical and the classical Thierrin radical of a graded ring, which is the direct sum of a family of its additive subgroups indexed by a nonempty set, under the assumption that the product of homogeneous elements is again homogeneous. There are two versions of this graded radical, the graded Thierrin and the large graded Thierrin radical. We establish several characterizations of the graded Thierrin radical and prove that the largest homogeneous ideal contained in the classical Thierrin radical of a graded ring coincides with the large graded Thierrin radical of that ring.     

On graded nil clean rings

In this paper we introduce and study the notion of a graded (strongly) nil clean ring which is group graded. We also deal with extensions of graded (strongly) nil clean rings to graded matrix rings and to graded group rings. The question of when nil cleanness of the component, which corresponds to the neutral element of a group, implies graded nil cleanness of the whole graded ring is examined. Similar question is discussed in the case of groupoid graded rings as well.     

Categorically perfect rings

An essential monoreflection is constructed on the category Sp of semiprirne commutative rings with identity, which is not a ring-of-quotients functor     

On prime ideals and radicals of polynomial rings and graded rings

We extend some known results on radicals and prime ideals from polynomial rings and Laurent polynomial rings to Z$\mathbb{Z}$-graded rings, i.e, rings graded by the additive group of integers. The main of them concerns the Brown–McCoy radical G$G$ and the radical S$S$, which for a given ring A$A$ is defined as the intersection of prime ideals I$I$ of A$A$ such that A/I$A/I$ is a ring with a large center. The studies are related to some open problems on the radicals G$G$ and S$S$ of polynomial rings and situated in the context of Koethe’s problem.     

Group graded rings

Let R be a group graded ring . The map ( , ): R × R →₁ defined by: (x,y) = (xy)₁ , is an inner product on R. In this paper we investigate aspects of nondegeneracy of the product, which is a generalization of the notion of strongly G —graded rings,introduced by Dade. We show that various chain conditions are satisfied by R if and only if they are satisfied by R₁ , and that when R₁ is simple artinian, then R is a crossed product R₁ ^* G. We give conditions for simple R-modules to be completely reducible R₁ -modules . Finally, we prove an incomparability theorem,when G is finite abelian.     

Almost perfect commutative rings

Almost perfect commutative rings R are introduced (as an analogue of Bazzoni and Salce's almost perfect domains) for rings with divisors of zero: they are defined as orders in commutative perfect rings such that the factor rings $R/Rr$ are perfect rings (in the sense of Bass) for all non-zero-divisors$r\in R$. It is shown that an almost perfect ring is an extension of a T-nilpotent ideal by a subdirect product of a finite number of almost perfect domains. Noetherian almost perfect rings are exactly the one-dimensional Cohen–Macaulay rings. Several characterizations of almost perfect domains carry over practically without change to almost perfect rings. Examples of almost perfect rings with zero-divisors are abundant.