Definition 1.1 (Rings). The algebraic structure “ring” R is a set with two binary operations + and ·, respectively named addition and multiplication, satisfying • (R,+) is an abelian group (i.e. a group with commutative addition), • is associative (i.e. 8a, b, c 2 R, (a · b) · c = a · (b · c)) , • and the distributive law holds (i.e. 8a, b, c 2 R, (a+ b) · c = a · c+ b · c, a · (b+ c) = a · b+ ...

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...

It is proved that a commutative ring is clean if and only if it is Gelfand with a totally disconnected maximal spectrum. It is shown that each indecomposable module over a commutative ring R satisfies a finite condition if and only if R P is an artinian valuation ring for each maximal prime ideal P. Commutative rings for which each indecomposable module has a local endomorphism ring are studied...

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.

let r be a right gf-closed ring with finite left and right gorenstein global dimension. we prove that if i is an ideal of r such that r/i is a semi-simple ring, then the gorensntein flat dimensnion of r/i as a right r-module and the gorensntein injective dimensnnion of r/i as a left r-module are identical. in particular, we show that for a simple module s over a commutative gorensntein ring r, ...

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...

Let $R$ be a commutative ring with identity and let $M$ be an $R$-module. In this paper we introduce a new graph associated to modules over commutative rings. We study the relationship between the algebraic properties of modules and their associated graphs. A topological characterization for the completeness of the special subgraphs is presented. Also modules whose associated graph is complete...

Let R be a ring. Let σ be an automorphism of R. We define a σ-divided ring and prove the following. (1) Let R be a commutative pseudovaluation ring such that x ∈ P for any P ∈ Spec(R[x,σ]) . Then R[x,σ] is also a pseudovaluation ring. (2) Let R be a σ-divided ring such that x ∈ P for any P ∈ Spec(R[x,σ]). Then R[x,σ] is also a σ-divided ring. Let now R be a commutative Noetherian Q-algebra (Q i...

Let B be a strictly real commutative real Banach algebra with the carrier space Φ B. If A is a commutative real Banach algebra, then we give a representation of a ring homomorphism ρ : A → B, which needs not be linear nor continuous. If A is a commutative complex Banach algebra, then ρ(A) is contained in the radical of B. 1. Introduction and results. Ring homomorphisms are mappings between two ...