in this paper we prove that each element of any regular baer ring is a sum of two units if no factor ring of r is isomorphic to z_2 and we characterize regular baer rings with unit sum numbers $omega$ and $infty$. then as an application, we discuss the unit sum number of some classes of group rings.

Let C be a coalgebra over a QF ring R. A left C-comodule is called strongly rational if its injective hull embeds in the dual of a right Ccomodule. Using this notion a number of characterizations of right semiperfect coalgebras over QF rings are given, e.g., C is right semiperfect if and only if C is strongly rational as left C-comodule. Applying these results we show that a Hopf algebra H over...

A ring $R$ is a strongly clean ring if every element in $R$ is the sum of an idempotent and a unit that commutate. We construct some classes of strongly clean rings which have stable range one. It is shown that such cleanness of $2 imes 2$ matrices over commutative local rings is completely determined in terms of solvability of quadratic equations.

Let R be a commutative ring. An R-module M is called co-multiplication provided that foreach submodule N of M there exists an ideal I of R such that N = (0 : I). In this paper weshow that co-multiplication modules are a generalization of strongly duo modules. Uniserialmodules of finite length and hence valuation Artinian rings are some distinguished classes ofco-multiplication rings. In additio...

An ideal I of a ring R is called right Baer-ideal if there exists an idempotent e 2 R such that r(I) = eR. We know that R is quasi-Baer if every ideal of R is a right Baer-ideal, R is n-generalized right quasi-Baer if for each I E R the ideal In is right Baer-ideal, and R is right principaly quasi-Baer if every principal right ideal of R is a right Baer-ideal. Therefore the concept of Baer idea...

If for any maximal right ideal P of B and a ∈ N(B) ,aB/ aP is almost N-injective, then ring said to be generalized N-injective. In this article, we present some significant findings that are known N-injective rings demonstrate they hold rings. At the same time, study case in which every S.S.Right B-module

‎A ring $R$ is strongly clean provided that every element‎ ‎in $R$ is the sum of an idempotent and a unit that commutate‎. ‎Let‎ ‎$T_n(R,sigma)$ be the skew triangular matrix ring over a local‎ ‎ring $R$ where $sigma$ is an endomorphism of $R$‎. ‎We show that‎ ‎$T_2(R,sigma)$ is strongly clean if and only if for any $ain‎ ‎1+J(R)‎, ‎bin J(R)$‎, ‎$l_a-r_{sigma(b)}‎: ‎Rto R$ is surjective‎. ‎Furt...

let r be a right gf-closed ring with finite left and right gorenstein global dimension. we prove that if i is an ideal of r such that r/i is a semi-simple ring, then the gorensntein flat dimensnion of r/i as a right r-module and the gorensntein injective dimensnnion of r/i as a left r-module are identical. in particular, we show that for a simple module s over a commutative gorensntein ring r, ...

The Faith-Menal conjecture says that every strongly right Johns ring is QF . The conjecture is also equivalent to say every right noetherian left FP -injective ring is QF . In this short article, we show that the conjecture is true under the condition( a proper generalization of left CS condition) that every nonzero complement left ideal is not small( a left ideal I is called small if for every...