Finitary ideals of direct products in quantales

The notion of quantale, which designates a complete lattice equipped with an associative binary multiplication distributing over arbitrary joins, appears in various areasof mathematics-in quantaloid theory, in non classical logic as completion of the Lindebaum algebra, and in different representations of the spectrum of a C∗ algebra asmany-valued and non commutative topologies. To put it briefly, its importance is nolonger to be demonstrated. Quantales are ring-like structures in that they share withrings the common fact that while as rings are semi groups in the tensor category ofabelian groups, so quantales are semi groups in the tensor category of sup-lattices.In 2008 Anderson and Kintzinger [1] investigated the ideals, prime ideals, radical ideals, primary ideals, and maximal of a product ring R × S of two commutativenon non necceray unital rings R and S: Something resembling rings are quantales byanalogy with what is studied in ring, we begin an investigation on ideals of a productof two quantales. In this paper, given two quantales Q1 and Q2; not necessarily withidentity, we investigate the ideals, prime ideals, primary ideals, and maximal ideals ofthe quantale Q1 × Q2

On some orders in ∗-rings based on the core-EP decomposition

We study certain relations in unital rings with involution that are derived from the core-EP decomposition. The notion of the WG pre-order and the C-E partial order is extended from [Formula: see text], the set of all [Formula: see text] matrices over [Formula: see text], to the set [Formula: see text] of all core-EP invertible elements in an arbitrary unital ring [Formula: see text] with involution. A new partial order is introduced on [Formula: see text] by combining the WG pre-order and the well known minus partial order, and a new characterization of the core-EP pre-order in unital proper ∗-rings is presented. Properties of these relations are investigated and some known results are thus generalized.

Étale groupoid algebras with coefficients in a sheaf and skew inverse semigroup rings

Given an action ${\varphi }$ of inverse semigroup S on a ring A (with domain of ${\varphi }(s)$ denoted by $D_{s^*}$ ), we show that if the ideals $D_e$ , with e an idempotent, are unital, then the skew inverse semigroup ring $A\rtimes S$ can be realized as the convolution algebra of an ample groupoid with coefficients in a sheaf of (unital) rings. Conversely, we show that the convolution algebra of an ample groupoid with coefficients in a sheaf of rings is isomorphic to a skew inverse semigroup ring of this sort. We recover known results in the literature for Steinberg algebras over a field as special cases.

Categorial properties of compressed zero-divisor graphs of finite commutative rings

By modifying the existing definition of a compressed zero-divisor graph [Formula: see text], we define a compressed zero-divisor graph [Formula: see text] of a finite commutative unital ring [Formula: see text], where the compression is performed by means of the associatedness relation (a refinement of the relation used in the definition of [Formula: see text]). We prove that this is the best possible compression which induces a functor [Formula: see text], and that this functor preserves categorial products (in both directions). We use the structure of [Formula: see text] to characterize important classes of finite commutative unital rings, such as local rings and principal ideal rings.

Some new characterizations of periodic rings

A ring [Formula: see text] is called periodic if, for every [Formula: see text] in [Formula: see text], there exist two distinct positive integers [Formula: see text] and [Formula: see text] such that [Formula: see text]. The paper is devoted to a comprehensive study of the periodicity of arbitrary unital rings. Some new characterizations of periodic rings and their relationship with strongly [Formula: see text]-regular rings are provided as well as, furthermore, an application of the obtained main results to a ∗-version of a periodic ring is being considered. Our theorems somewhat considerably improved on classical results in this direction.