Let $R$ be a commutative Noetherian ring. We prove that  over a local ring $R$ every finitely generated $R$-module $M$ of finite Gorenstein projective dimension has a Gorenstein projective cover$varphi:C rightarrow M$ such that $C$ is finitely generated and the projective dimension of $Kervarphi$ is finite and $varphi$ is surjective.

There is a local ring [Formula: see text] of order without identity for the multiplication, defined by generators and relations as We study recursive construction self-orthogonal codes over classify, up to permutation equivalence, length size (called here quasi self-dual or QSD) text]. In particular, we classify Type IV (QSD with even weights)

General Relativity differs from most classical field theories in that there is no welldefined notion of energy density for the gravitation field, as can be seen from Einstein’s principle of equivalence. Thus at best one can only hope to calculate the mass/energy contained within a domain, as opposed to at a point. Such a concept is referred to as quasi-local mass, that is, a functional which as...

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.

 module over a ring is a general mathematical concept for many examples of mathematicalobjects that can be added to each other and multiplied by scalar numbers.In this paper, we consider a module over a ring as a universe and by using the notion of reference points, we provide local approximations for  subsets of the universe.

Let $(R,fm,k)$ be a local Gorenstein ring of dimension $n$. Let $H_{I,J}^i(R)$ be the  local cohomology with respect to a pair of ideals $I,J$ and $c$ be the $inf{i|H_{I,J}^i(R)neq0}$. A pair of ideals $I, J$ is called cohomologically complete intersection if $H_{I,J}^i(R)=0$ for all $ineq c$. It is shown that, when $H_{I,J}^i(R)=0$ for all $ineq c$, (i) a minimal injective resolution of $H_{I,...