for an arbitrary ring $r$, the zero-divisor graph of $r$, denoted by $gamma (r)$, is an undirected simple graph that its vertices are all nonzero zero-divisors of $r$ in which any two vertices $x$ and $y$ are adjacent if and only if either $xy=0$ or $yx=0$. it is well-known that for any commutative ring $r$, $gamma (r) cong gamma (t(r))$ where $t(r)$ is the (total) quotient ring of $r$. in this...

The structure of arbitrary associative commutative unital artinian algebras is well-known: they are finite products of associative commutative unital local algebras [6, pg.351, Cor. 23.12]. In the semi-simple case, we have the Artin-Wedderburn Theorem which states that any semi-simple artinian algebra (which is assumed to be associative and unital but not necessarily commutative) is a direct pr...

For an arbitrary left Artinian ring R, explicit descriptions are given of all the left denominator sets S of R and left localizations SR of R. It is proved that, up to R-isomorphism, there are only finitely many left localizations and each of them is an idempotent localization, i.e. SR ≃ S e R and ass(S) = ass(Se) where Se = {1, e} is a left denominator set of R and e is an idempotent. Moreover...

Let R be a (mixed characteristic) Artinian local ring of length l and let X be an n-tuple of variables. This paper provides bounds over the ring R[X] on the degrees of the output of several algebraic constructions in terms of l, n and the degrees of the input. For instance, if I is an ideal in R[X] generated by polynomials gi of degree at most d and if f is a polynomial of degree at most d belo...

The purpose of this paper is to provide a criterion of an occurrence of uncountably generated uniserial modules over chain rings. As we show it suffices to investigate two extreme cases, nearly simple chain rings, i.e. chain rings containing only three twosided ideals, and chain rings with “many” two-sided ideals. We prove that there exists an ω1-generated uniserial module over every non-artini...

We apply minimal weakly generating sets to study the existence of Add ( U R ) -covers for a uniserial module . If is right over ring , then S : = End has at most two maximal (right, left, two-sided) ideals: one set I all endomorphisms that are not injective, and other K surjective. prove if either finitely generated, or artinian, ? class covering only it closed under direct limit. Moreover, we ...

It is proved that a ring A right or left Noetherian, distributive, centrally essential if and only [Formula: see text], where each of the rings text] either commutative Dedekind domain Artinian, uniserial ring.

we introduce the class of “right almost v-rings” which is properly between the classes of right v-rings and right good rings. a ring r is called a right almost v-ring if every simple r-module is almost injective. it is proved that r is a right almost v-ring if and only if for every r-module m, any complement of every simple submodule of m is a direct summand. moreover, r is a right almost v-rin...