Narrow Results

A ring $R$ is said to be right $\pi$-extending if every projection invariant right ideal of $R$ is essential in a direct summand of $R$. In this article, we investigate the transfer of the $\pi$-extending condition between a ring $R$ and its various ring extensions. More specifically, we characterize the right $\pi$-extending generalized triangular matrix rings; and we show that if $R_R$ is $\pi$-extending, then so is $T_T$ where $T$ is an overring of $R$ which is an essential extension of $R$, an $n\times n$ upper triangular matrix ring of $R$, a column finite or column and row finite matrix ring over $R$, or a certain type of trivial extension of $R$.

Let ${\mathbb{Z}}_{p^s}$ be the ring of integers modulo $p^s$ where $p$ is a prime and $s(\ge 1)$ is a positive integer, $R=M_{2\times 2}({\mathbb{Z}}_{p^s})$ the $2\times 2$ matrix ring over ${\mathbb{Z}}_{p^s}$. The zero-divisor graph of $R$, written as $\mathrm{\Gamma }(R)$, is a directed graph whose vertices are nonzero zero-divisors of $R$, and there is a directed edge from a vertex $A$ to a vertex $B$ if and only if $AB=0$. In this paper, we completely determine the automorphisms of $\mathrm{\Gamma }(R)$.

As introduced by Cǎlugǎreanu and Lam in [G. Cǎlugǎreanu and T. Y. Lam, Fine rings: a new class of simple rings, J. Algebra Appl.15(9) (2016) 1650173, 18 pp.], a fine ring is a ring whose every nonzero element is the sum of a unit and a nilpotent. As a natural generalization of fine rings, a ring is called a generalized fine ring if every element not in the Jacobson radical is the sum of a unit and a nilpotent. Here some known results on fine rings are extended to generalized fine rings. A notable result states that matrix rings over generalized fine rings are generalized fine, extending the important result in [G. Cǎlugǎreanu and T. Y. Lam, Fine rings: a new class of simple rings, J. Algebra Appl.15(9) (2016) 1650173, 18 pp.] that matrix rings over fine rings are fine.

Let $R$ be a commutative ring with identity, $n\ge 2$ be a positive integer and $M_n(R)$ be the set of all $n\times n$ matrices over $R.$ For a matrix $A\in M_n(R),$ Tr$(A)$ is the trace of $A.$ The trace graph of the matrix ring $M_n(R),$ denoted by ${\mathrm{\Gamma }}_t(M_n(R)),$ is the simple undirected graph with vertex set $\{A\in M_n{(R)}^{\ast }:\text{there exists }B\in M_n{(R)}^{\ast }$$\text{ such that Tr}(AB)=0\}$ and two distinct vertices $A$ and $B$ are adjacent if and only if Tr$(AB)=0.$ The ideal-based trace graph of the matrix ring $M_n(R)$ with respect to an ideal $I$ of $R,$ denoted by ${\mathrm{\Gamma }}_{I^t}(M_n(R)),$ is the simple undirected graph with vertex set $M_n(R)\setminus M_n(I)$ and two distinct vertices $A$ and $B$ are adjacent if and only if Tr$(AB)\in I.$ In this paper, we investigate some properties and structure of ${\mathrm{\Gamma }}_{I^t}(M_n(R)).$ Further, it is proved that both ${\mathrm{\Gamma }}_t(M_n(R))$ and ${\mathrm{\Gamma }}_{I^t}(M_n(R))$ are Hamiltonian.

In this paper, we introduce and investigate three new versions of the Rickart condition for rings. These conditions, as well as, three new corresponding regularities are defined using projection invariance. We show how these conditions relate to each other as well as their connections to the well-known Baer, Rickart, quasi-Baer, p.q.-Baer, regular, and biregular conditions. Applications to polynomial extensions and to triangular and full matrix rings are provided. Examples illustrate and delimit results.

This paper concerns several ring theoretic properties related to matrices and polynomials. The basic properties of $\pi$-reversible and power-Armendariz are studied. We provide a method by which one can always construct a power-Armendariz ring but neither symmetric nor Armendariz from given any symmetric ring. We investigate next various interesting relations among ring theoretic properties containing $\pi$-reversibility and power-Armendariz condition.

We introduce the set S(R) of "strong zero-divisors" in a ring R and prove that: if S(R) is finite, then R is either finite or a prime ring. When certain sets of ideals have ACC or DCC, we show that either S(R) = R or S(R) is a union of prime ideals each of which is a left or a right annihilator of a cyclic ideal. This is a finite union when R is a Noetherian ring. For a ring R with |S(R)| = p, a prime number, we characterize R for S(R) to be an ideal. Moreover R is completely characterized when R is a ring with identity and S(R) is an ideal with p² elements. We then consider rings R for which S(R)= Z(R), the set of zero-divisors, and determine strong zero-divisors of matrix rings over commutative rings with identity.

A ring R is called pseudopolar if for every a ∈ R there exists p² = p ∈ R such that p ∈ comm²(a), a + p ∈ U(R) and a^kp ∈ J(R) for some positive integer k. Pseudopolar rings are closely related to strongly π-regular rings, uniquely strongly clean rings, semiregular rings and strongly π-rad clean rings. In this paper, we completely characterize the local rings R for which M₂(R) is pseudopolar.

We in this note consider the reflexive ring property on nil ideals, introducing the concept of a nil-reflexive ring as a generalization of the reflexive ring property. We will call a ring $R$nil-reflexive if $IJ=0$ implies $JI=0$ for nil ideals $I,J$ of $R$. The polynomial and the power series rings over a right Noetherian ring (or an NI ring) $R$ are shown to be nil-reflexive if ${(aRb)}^2=0$ implies $aRb=0$ for all $a,b\in N(R)$. We further investigate the structure of nil-reflexive rings, related to various sorts of ring extensions which have roles in ring theory.

Mason introduced the reflexive property for ideals, and recently this concept was extended to many sorts of subsets in rings. In this note, we restrict the reflexivity to nilpotent elements, and a ring will be said to be RNP if it satisfies this restriction. The structure of RNP rings is studied in relation to the near concepts and ring extensions which have roles in ring theory.

Let $F_q$ be a finite field with $q$ elements, $R_0=M_n(F_q)$ be the ring of all $n\times n$ matrices over $F_q$, $L(R_0)$ be the set of all nontrivial left ideals of $R_0$. The co-maximal ideal graph of $R_0$, denoted by $C(R_0)$, is a graph with $L(R_0)$ as vertex set and two nontrivial left ideals $I,J$ of $R_0$ are adjacent if and only if $I+J=R_0$. If $n=2$, it is easy to see that $C(R_0)$ is a complete graph, thus any permutation of vertices of $C(R_0)$ is an automorphism of $C(R_0)$. A natural problem is: How about the automorphisms of $C(R_0)$ when $n\ge 3$. In this paper, we aim to solve this problem. When $n\ge 3$, a mapping $\sigma$ on $L(R_0)$ is proved to be an automorphism of $C(R_0)$ if and only if there is an invertible matrix $x\in R_0$ and a field automorphism $f$ of $F_q$ such that $\sigma (I)=f(I)x$ for any $I\in L(R_0)$, where $f(I)x=\{f(z)x|z\in I\}$ and $f(z)={[f(z_{ij})]}_{n\times n}$ for $z={[z_{ij}]}_{n\times n}\in R_0$.

Let $R = \left(\begin{array}{cc}A& M\\ N& B\end{array}\right)$ be a Morita context. For generalized fine (respectively, generalized unit-fine) rings $A$ and $B$, it is proved that $R$ is generalized fine (respectively, generalized unit-fine) if and only if, for $a\in A$ and $b\in B$, $MbN\subseteq J(A)$ implies $b\in J(B)$ and $NaM\subseteq J(B)$ implies $a\in J(A)$. Especially, for fine (respectively, unit-fine) rings $A$ and $B$, $R$ is fine (respectively, unit-fine) if and only if, for $a\in A$ and $b\in B$, $MbN=0$ implies $b=0$ and $NaM=0$ implies $a=0$. As consequences, (1) matrix rings over fine (respectively, unit-fine, generalized fine and generalized unit-fine) rings are fine (respectively, unit-fine, generalized fine and generalized unit-fine); (2) a sufficient condition for a simple ring to be fine (respectively, unit-fine) is obtained: a simple ring $R$ is fine (respectively, unit-fine) if both $eRe$ and $(1-e)R(1-e)$ are fine (respectively, unit-fine) for some $e^2=e\in R$; and (3) a question of Cǎlugǎreanu [1] on unit-fine matrix rings is affirmatively answered.

If $R$ is a ring then the square element graph $𝕊q(R)$ is the simple undirected graph whose vertex set consists of all non-zero elements of $R$ and two distinct vertices $u,v$ are adjacent if and only if $u+v=x^2$ for some $x\in R\setminus \{0\}$. In this paper, we provide some necessary and sufficient conditions for the connectedness of $𝕊q(R)$, where $R$ is a ring with identity. We mainly characterize some special class of ring $R$ which we call square-subtract ring for which the graph $𝕊q(M_n(R))$ is connected.