Centrally Prime Rings which are Commutative (original) (raw)
On Centrally Prime and Centrally Semiprime Rings
AL-Rafidain Journal of Computer Sciences and Mathematics, 2008
In this paper, centrally prime and centrally semiprime rings are defined and the relations between these two rings and prime (resp. semiprime) rings are studied.Among the results of the paper some conditions are given under which prime (resp. semiprime) rings become centrally prime (resp.centrally semiprime) as in:1-A nonzero prime (resp. semiprime) ring which has no proper zero divisors is centrally prime (resp.centrally semiprime).Also we gave some other conditions which make prime (resp. semiprime) rings and centrally prime (resp.centrally semiprime) rings equivalent, as in :2-A ring which satisfies the-) (BZP for multiplicative systems is prime (resp. semiprime) if and only if it is centrally prime (resp.centrally semiprime).3-A ring with identity in which every nonzero element of its center is a unit is prime (resp. semiprime) if and only if it is centrally prime (resp.centrally semiprime).
International Journal of Mathematics and Mathematical Sciences, 1994
LetRbe a ring, and letCdenote the center ofR.Ris said to have a prime center if wheneverabbelongs toCthenabelongs toCorbbelongs toC. The structure of certain classes of these rings is studied, along with the relation of the notion of prime centers to commutativity. An example of a non-commutative ring with a prime center is given.
On Centrally Semiprime Rings and Centrally Semiprime
Maǧallaẗ ǧāmiʻaẗ kirkūk, 2008
In this paper, two new algebraic structures are introduced which we call a centrally semiprime ring and a centrally semiprime right near-ring, and we look for those conditions which make centrally semiprime rings as commutative rings, so that several results are proved, also we extend some properties of semiprime rings and semiprime right near-rings to centrally semiprime rings and centrally semiprime right near-rings.
Some Properties of Centrally Closed Lie Ideals of Centrally Prime Rings
Journal of Al-Nahrain University-Science, 2007
In this paper, the definitions of a centrally closed Lie ideal and central identity property are introduced and the behaviors of this type of ideals in 2-torsion free centrally prime rings are studied, also we study the effects of derivation on them, so we will prove some properties of these ideals.
2015
Dedicated to S. K. Jain in honor of his 70th birthday. Abstract. We establish commutativity theorems for certain classes of rings in which every invertible element is central, or, more generally, in which all invertible elements commute with one another. We prove that if R is a semiex-change ring (i.e. its factor ring modulo its Jacobson radical is an exchange ring) with all invertible elements central, then R is commutative. We also prove that if R is a semiexchange ring in which all invertible elements com-mute with one another, and R has no factor ring with two elements, then R is commutative. We offer some examples of noncommutative rings in which all invertible elements commute with one another, or are central. We close with a list of problems for further research.
Commutativity for a certain class of rings
Georgian Mathematical Journal, 1996
We first establish the commutativity for the semiprime ring satisfying [x n , y]x r = ±y s [x, y m ]y t for all x, y in R, where m, n, r, s and t are fixed non-negative integers, and further, we investigate the commutativity of rings with unity under some additional hypotheses. Moreover, it is also shown that the above result is true for s-unital rings. Also, we provide some counterexamples which show that the hypotheses of our theorems are not altogether superfluous. The results of this paper generalize some of the well-known commutativity theorems for rings which are right s-unital.
A Commutativity theorem for semiprime rings
Journal of the Australian Mathematical Society, 1980
It is shown that if R is a semiprime ring with 1 satisfying the property that, for each x, y e R, there exists a positive integer n depending on v and y such that (\_v)*-x*>'*is central for k = n,n+ 1,H + 2, then R is commutative, thus generalizing a result of Kaya.
On strongly prime rings and ideals
Communications in Algebra, 2000
Strongly prime rings may be defined as prime rings with simple central closure. This paper is concerned with further investigation of such rings. Various characterizations, particularly in terms of symmetric zero divisors, are given. We prove that the central closure of a strongly (semi-)prime ring may be obtained by a certain symmetric perfect one sided localization. Complements of strongly prime ideals are described in terms of strongly multiplicative sets of rings. Moreover, some relations between a ring and its multiplication ring are examined.
2016
Let R be a *-prime ring with center Z(R), d a non-zero (σ, τ)-derivation of R with associated automorphisms σ and τ of R, such that σ, τ and d commute with ′ * ′. Suppose that U is an ideal of R such that U * = U , and Cσ,τ = {c ∈ R | cσ(x) = τ (x)c for all x ∈ R}. In the present paper, it is shown that if characteristic of R is different from two and [d(U), d(U)]σ,τ = {0}, then R is commutative. Commutativity of R has also been established in case if [d(R), d(R)]σ,τ ⊆ Cσ,τ .
Characterization of prime rings having involution and centralizers
Proyecciones, 2023
The major goal of this paper is to study the commutativity of prime rings with involution that meet specific identities using left centralizers. The results obtained in this paper are the generalization of many known theorems. Finally, we provide some examples to show that the conditions imposed in the hypothesis of our results are not superfluous.