Subgraph of generalized co-maximal graph of commutative rings

Let R be a commutative ring with 1. In [3], we introduced a graph G(R) whose vertices are elements of R and two distinct vertices a, b are adjacent if and only if aR + bR = eR for some non-zero idempotent e in R. Let G′(R) be the subgraph of G(R) generated by the non-units of R. In this paper, we characterize those rings R for which the graph G′(R) is connected and Eulerian. Also we characterize those rings R for which genus of the graph G′(R) is ≤ 2. Finally, we show that the graph G′(R) is a line graph of some graph if and only if R is either a regular ring or a local ring.AMS Subject Classification 2020 : 05C25

A Property Satisfying Reducedness over Centers

This article concerns a ring property called pseudo-reduced-over-center that is satisfied by free algebras over commutative reduced rings. The properties of radicals of pseudo-reduced-over-center rings are investigated, especially related to polynomial rings. It is proved that for pseudo-reduced-over-center rings of nonzero characteristic, the centers and the pseudo-reduced-over-center property are preserved through factor rings modulo nil ideals. For a locally finite ring [Formula: see text], it is proved that if [Formula: see text] is pseudo-reduced-over-center, then [Formula: see text] is commutative and [Formula: see text] is a commutative regular ring with [Formula: see text] nil, where [Formula: see text] is the Jacobson radical of [Formula: see text].

Commutative von Neumann regular rings are 1-Gröbner

Let R be a commutative von Neumann regular ring. We show that every finitely generated ideal I in the ring of polynomials R[X] has a strong Gr?bner basis. We prove this result using only the defining property of a von Neumann regular ring.

A note on a generalization of injective modules

As a proper generalization of injective modules in term of supplements, we say that a module $M$ has the property (ME) if, whenever $M\subseteq N$, $M$ has a supplement $K$ in $N$, where $K$ has a mutual supplement in $N$. In this study, we obtain that $(1)$ a semisimple $R$-module $M$ has the property (E) if and only if $M$ has the property (ME); $(2)$ a semisimple left $R$-module $M$ over a commutative Noetherian ring $R$ has the property (ME) if and only if $M$ is algebraically compact if and only if almost all isotopic components of $M$ are zero; $(3)$ a module $M$ over a von Neumann regular ring has the property (ME) if and only if it is injective; $(4)$ a principal ideal domain $R$ is left perfect if every free left $R$-module has the property (ME)

Varieties of ∗-regular rings

Given a subdirectly irreducible ∗-regular ring R, we show that R is a homomorphic image of a regular ∗-subring of an ultraproduct of the (simple) eRe, e in the minimal ideal of R; moreover, R (with unit) is directly finite if all eRe are unit-regular. For any subdirect product of artinian ∗-regular rings we construct a unit-regular and ∗-clean extension within its variety.

A Theorem About Maximal Cohen–Macaulay Modules

It is shown that for any local strongly $F$-regular ring there exists natural number $e_0$ so that if $M$ is any finitely generated maximal Cohen–Macaulay module, then the pushforward of $M$ under the $e_0$th iterate of the Frobenius endomorphism contains a free summand. Consequently, the torsion subgroup of the divisor class group of a local strongly $F$-regular ring is finite.

On P Wπ-regular rings

As a popularization of weakly π-regular rings, we tender the connotation of W P Wπ-regular rings, that is if for each 𝔞 ∈ Ɉ(𝔑), there exist a natural number 𝔫 such that 𝔞𝔫 ∈ 𝔞𝔫 𝔑 𝔞𝔫 𝔑 (𝔞𝔫 ∈ 𝔑 𝔞𝔫 𝔑 𝔞𝔫). In this treatise, numerous properties of this sort of rings are discussed, some important results are secured. Using the connotation of P Wπ-regular rings. It is show that : 1- Let 𝔑 be a right P Wπ-regular ring and ℵɈ-rings with 𝔞𝔫𝔑 = 𝔑𝔞𝔫 for every 𝔞 ∈ Ɉ(𝔑) and for at least one of a natural number. Then Ɉ(𝔑) = ℵ𝔑. 2- Let 𝔑 a right P Wπ-regular ring and 𝔞𝔑 = 𝔑𝔞 for each ∈Ɉ(𝔑). Then 𝔑 is right P .𝔗-ring. 3 Let 𝔑 be a ring with ɍ(𝔞) ⊆ ɭ(𝔞), for each ∈ Ɉ (𝔑). If any of the next conditions are hold, then 𝔑 is P Wπ-regular rings : i – Every maximal right ideal of 𝔑 is a right annihilator and right Ɉ PP -ring. ii – any simple singular right 𝔑-module is Ɉ-injective and 𝔑 is semi prime.