Let R be a commutative ring with identity and T (R) its total quotient ring. We extend the notion of well-centered overring of an integral domain to an arbitrary commutative ring and we investigate the transfer of this property to different extensions of commutative rings in both integral and non-integral cases. Namely in pullbacks and trivial extensions. Our aim is to provide new classes of co...

We begin with some reminders and remarks on non-commutative rings. First, recall that if R is a not necessarily commutative ring, an element x ∈ R is invertible if it has both a left (multiplicative) inverse and a right (multiplicative) inverse; it then follows that the two are necessarily equal. Note however, that unlike the example of n× n matrices over a field, in a general noncommutative ri...

We give a general formula relating self cup products in cohomology to connecting maps in nonabelian cohomology, and apply it to obtain a formula for the self cup product associated to the Weil pairing.

Prime graph of a ring R is a graph whose vertex set is the whole set R any any two elements $x$ and $y$ of $R$ are adjacent in the graph if and only if $xRy = 0$ or $yRx = 0$.  Prime graph of a ring is denoted by $PG(R)$. Directed prime graphs for non-commutative rings and connectivity in the graph are studied in the present paper. The diameter and girth of this graph are also studied in the pa...

In this paper, we propose a new method for designing public key cryptosystems based on general non-commutative rings. The key idea of our proposal is that for a given non-commutative ring, we can define polynomials and take them as the underlying work structure. By doing so, it is easy to implement Diffie-Helman-like key exchange protocol. And consequently, ElGamal-like cryptosystems can be der...

the present note arises from the author's talk at the conference ``ischia group theory 2014''. for subgroups $fle n$ of a group $g$ denote by $lat(f,n)$ the set of all subgroups of $n$, containing $f$. let $d$ be a subgroup of $g$. in this note we study the lattice $ll=lat(d,g)$ and the lattice $ll'$ of subgroups of $g$, normalized by $d$. we say that $ll$ satisfies sandwich classification theo...

We study the properties of rings satisfying Auslander-type conditions. If an artin algebra Λ satisfies the Auslander condition (that is, Λ is an ∞-Gorenstein artin algebra), then we construct two kinds of subcategories which form functorially finite cotorsion theories. Noetherian rings satisfying ‘Auslander-type conditions’ on self-injective resolutions can be regarded as certain non-commutativ...

the aim of this paper is to generalize the‎‎notion of pseudo-almost valuation domains to arbitrary‎ ‎commutative rings‎. ‎it is shown that the classes of chained rings‎ ‎and pseudo-valuation rings are properly contained in the class of‎ ‎pseudo-almost valuation rings; also the class of pseudo-almost‎ ‎valuation rings is properly contained in the class of quasi-local‎ ‎rings with linearly ordere...

Let $R$ be a commutative ring with identity. An ideal $I$ of a ring $R$is called an annihilating ideal if there exists $rin Rsetminus {0}$ such that $Ir=(0)$ and an ideal $I$ of$R$ is called an essential ideal if $I$ has non-zero intersectionwith every other non-zero ideal of $R$. Thesum-annihilating essential ideal graph of $R$, denoted by $mathcal{AE}_R$, isa graph whose vertex set is the set...