Narrow Results

This paper concerns ring properties which are induced from the structure of the powers of prime ideals. An ideal $I$ of a ring $R$ is called $n$-primary (respectively, $T$-primary) provided that $AB\subseteq I$ for ideals $A,B$ of $R$ implies that $(A+I)/I$ or $(B+I)/I$ is nil of index $n$ (respectively, $(A+I)/I$ or $(B+I)/I$ is nil) in $R/I$, where $n\ge 1$. It is proved that for a proper ideal $I$ of a principal ideal domain $R$, $I$ is $T$-primary if and only if $I$ is of the form $p^{k}R$ for some prime element $p$ and $k\ge 1$ if and only if $I$ is $2$-primary, through which we study the structure of matrices over principal ideal domains. We prove that for a $T$-primary ideal $I$ of a ring $R$, $R/I$ is prime when the Wedderburn radical of $R/I$ is zero. In addition we provide a method of constructing strictly descending chain of $n$-primary radicals from any domain, where the $n$-primary radical of a ring $R$ means the intersection of all the $n$-primary ideals of $R$.

For a nonempty subset $X$ of a ring $R$, the ring $R$ is called $X$-semiprime if, given $a\in R$, $aXa=0$ implies $a=0$. This provides a proper class of semiprime rings. First, we clarify the relationship between idempotent semiprime and unit-semiprime rings. Secondly, given a Lie ideal $L$ of a ring $R$, we offer a criterion for $R$ to be $L$-semiprime. For a prime ring $R$, we characterizes Lie ideals $L$ of $R$ such that $R$ is $L$-semiprime. Moreover, $X$-semiprimeness of matrix rings, prime rings (with a nontrivial idempotent), semiprime rings, regular rings, and subdirect products are studied.

Different public key exchange protocols can be employed to design a cryptosystem. The most famous example is the ElGamal cryptosystem which is based on the Diffie–Hellman key establishment to do encryption and decryption. In the same fashion, we propose a public key cryptosystem using a public key exchange protocol. This variant uses the polynomials over noncommutative matrix ring as underlying work structure. The useful feature of the proposed cryptosystem is that it provides favorable security because of use of the generalized decomposition problem over the noncommutative ring of matrices. The issues regarding the choice of parameters and platform are discussed. Further, a brief note on security analysis of the proposed cryptosystem is also presented.

Rings of matrices whose idempotents are closed under multiplication are studied and those rings that are maximal subject to certain constraints are characterized.

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.

In this paper, we deal with the problem of computing the sum of the $k$th powers of all the elements of the matrix ring ${𝕄}_d(R)$ with $d>1$ and $R$ a finite commutative ring. We completely solve the problem in the case $R=\mathbb{Z}/n\mathbb{Z}$ and give some results that compute the value of this sum if $R$ is an arbitrary finite commutative ring for many values of $k$ and $d$. Finally, based on computational evidence and using some technical results proved in this paper, we conjecture that the sum of the $k$th powers of all the elements of the matrix ring ${𝕄}_d(R)$ is always $0$ unless $d=2$, $\mathrm{\text{card}}(R)\equiv 2(\mathrm{\text{mod}}4)$, $1<k\equiv -1,0,1(\mathrm{\text{mod}}6)$ and the only element $e\in R\setminus \{0\}$ such that $2e=0$ is idempotent, in which case the sum is $\mathrm{\text{diag}}(e,e)$.

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.

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 $R$.

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.

An element in a ring is strongly nil-clean, if it is the sum of an idempotent and a nilpotent element that commute. A ring $R$ is strongly unit nil-clean, if for any $a\in R$ there exists a unit $u\in R$, such that $ua$ is strongly nil-clean. We prove, in this paper, that a ring $R$ is strongly unit nil-clean, if and only if every element in $R$ is equivalent to a strongly nil-clean element, if and only if for any $a\in R$, there exists a unit $u\in R$, such that $ua\in R$ is strongly $\pi$-regular. Strongly unit nil-clean matrix rings are investigated as well.

In this note, we focus our attention on a new ring structure related to annihilators, and consider a ring property that contains many kinds of ring classes, introducing right ZAFS. This property is shown to be not left-right symmetric but left-right symmetric for left or right Artinian rings. The left (right) ZAFS property is shown to pass to Ore extensions with automorphisms. The left (respectively, right) ZAFS property is shown to pass also to classical left (respectively, right) quotient rings, yielding that semiprime right Goldie rings are ZAFS.

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 ${\mathbb{Z}}_{p^s}$ 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 $\mathrm{\Gamma }(R)$.

The usual commutative ideal theory was extended to ideals in noncommutative rings by Lambek, introducing the concept of symmetric. Camillo et al. naturally extended the study of symmetric ring property to the lattice of ideals, defining the new concept of an ideal-symmetric ring. This paper focuses on the symmetric ring property on nil ideals, as a generalization of an ideal-symmetric ring. A ring $R$ will be said to be right (respectively, left) nil-ideal-symmetric if $IJK=0$ implies $IKJ=0$ (respectively, $JIK=0$) for nil ideals $I,J,K$ of $R$. This concept generalizes both ideal-symmetric rings and weak nil-symmetric rings in which the symmetric ring property has been observed in some restricted situations. The structure of nil-ideal-symmetric rings is studied in relation to the near concepts and ring extensions which have roles in ring theory.

The purpose of this paper is to provide useful connections between units and zero divisors, by investigating the structure of a class of rings in which Köthe’s conjecture (i.e. the sum of two nil left ideals is nil) holds. We introduce the concept of unit-IFP for the purpose, in relation with the inserting property of units at zero products. We first study the relation between unit-IFP rings and related ring properties in a kind of matrix rings which has roles in noncommutative ring theory. The Jacobson radical of the polynomial ring over a unit-IFP ring is shown to be nil. We also provide equivalent conditions to the commutativity via the unit-IFP of such matrix rings. We construct examples and counterexamples which are necessary to the naturally raised questions.

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.

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 $R = \left(\begin{array}{cc}A& M\\ N& B\end{array}\right)$ 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.