Throughout the paper, all rings are associative rings with identity 1. The set of all idempotents of a ring R is denoted by E(R). Also, for a subset X ⊆ R, we denote the right [resp., left] annihilator of X by r(X) [resp., (X)]. We call a ring R a left p.p.-ring [3], in brevity, an l.p.p.-ring, if every principal left ideal of R, regarded as a left R-module, is projective. Dually, we may define...

In this errata, we reconsider and modify two propositions and their corollaries which were written on epi-retractable and co-epi-retractable modules.

Hereditary benign telangiectasia is an uncommon idiopathic skin disorder characterized by generalized telangiectases and angiomatous lesions of the skin. The diagnosis should be suspected in patients with such cutaneous lesions, positive family history, no associated bleeding problems, and no mucosal involvement. We present six cases of hereditary benign telangiectasia in three generations of o...

let $r$ be a reversible ring which is $alpha$-compatible for an endomorphism $alpha$ of $r$ and $f(x)=a_0+a_1x+cdots+a_nx^n$ be a nonzero skew polynomial in $r[x;alpha]$. it is proved that if there exists a nonzero skew polynomial $g(x)=b_0+b_1x+cdots+b_mx^m$ in $r[x;alpha]$ such that $g(x)f(x)=c$ is a constant in $r$, then $b_0a_0=c$ and there exist nonzero elements $a$ and $r$ in $r$ such tha...

in this paper, we introduce the concept of the   generalized aip-rings as a generalization of the generalized quasi-   baer rings   and generalized p.p.-rings. we show that the class of the generalized aip-rings is closed under direct products and morita invariance. we also characterize the 2-by-2 formal upper triangular matrix rings of this new class of rings. finally, we provide sever...

We provide a unified approach, via deformations of incidence algebras, to several important types representations with finiteness conditions, as well the combinatorial algebras which produce them. show that over finite dimensional finitely many orbits, or invariant subspaces, distributive coincide, and further coincide thin modules in acyclic case. Incidence examples such modules, we are locall...

this paper is an attempt to generalize, simultaneously, the ring of real-valued continuous functions and the ring of real-valued measurable functions.

let $r$ be an associative ring with identity and $z^*(r)$ be its set of non-zero zero divisors.  the zero-divisor graph of $r$, denoted by $gamma(r)$, is the graph whose vertices are the non-zero  zero-divisors of  $r$, and two distinct vertices $r$ and $s$ are adjacent if and only if $rs=0$ or $sr=0$.  in this paper, we bring some results about undirected zero-divisor graph of a monoid ring ov...