THE SOURCE OF SEMIPRIMENESS OF 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.

2013

A well-known theorem of Wedderburn asserts that a finite division ring is commutative. In a division ring the group of invertible elements is as large as possible. Here we will be particularly interested in the case where this group is as small as possible, namely reduced to 1. We will show that, if this is the case, then the ring is boolean. Thus, here too, the ring is commutative. Classification: 16K99, 16N99 Notation We will write |S| for the cardinal of a set S. If R is a ring, then we will denote the subset of its nonzero elements R * and the subset of its invertible elements R ×. If the ring is a division ring, then R * = R ×. Also, we will write char(R) for the characteristic of a ring R.

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.

Canadian Journal of Mathematics, 1980

Throughout this paper the ring R and the semigroup S are commutative with identity; moreover, it is assumed that S is cancellative, i.e., that S can be embedded in a group. The aim of this note is to determine necessary and sufficient conditions on R and 5 that the semigroup ring R[S] should be one of the following types of rings: principal ideal ring (PIR), ZPI-ring, Bezout, semihereditary or arithmetical. These results shed some light on the structure of semigroup rings and provide a source of examples of the rings listed above. They also play a key role in the determination of all commutative reduced arithmetical semigroup rings (without the cancellative hypothesis on S) which will appear in a forthcoming paper by Leo Chouinard and the authors [4]. Our results are motivated in large part by the paper [11] of R. Gilmer and T. Parker. In particular, Theorem 1.1 of [11] asserts that if R and S are as above and, moreover, if 5 is torsion-free, then the following are equivalent conditions: (1) R[S] is a Bezout ring; (2) R[S] is a Priifer ring; (3) R is a (von Neumann) regular ring and 5 is isomorphic to either a subgroup of the additive rationals or the positive cone of such a subgroup. One could very naturally include a fourth condition, namely: (4) R[S] is arithmetical. L. Fuchs [7] defines an arithmetical ring as a commutative ring with identity for which the ideals form a distributive lattice. Since a Priifer ring is one for which (A + B) C\ C = {A C\ C) + (B Pi C) whenever at least one of the ideals A, B or C contains a regular element (see [18]), arithmetical rings are certainly Priifer. On the other hand, it is well known that every Bezout ring is arithmetical, so that (4) is indeed equivalent to (l)-(3) in Theorem 1.1. In Theorem 3.6 of this paper we drop the requirement that S be torsion-free and determine necessary and sufficient conditions for the semigroup ring of a cancellative semigroup to be arithmetical. Examples are included to show that for these more general semigroup rings, the equivalences of the torsion-free case are no longer true. Theorems 4.1 and 4.2 provide characterizations of semigroup rings that are ZPI-rings and PIR's. Again, the corresponding results in [18] for torsion-free semigroups fail to hold in the more general case. We would like to thank Leo Chouinard for showing us how to remove

Proceedings of the American Mathematical Society, 1973

The structure of prime rings has recently been studied by A

International Journal of Algebra

Let R be a semiprime ring. An additive mapping d : R → R is called a semiderivation if there exists a function g : R → R such that (i) d(xy) = d(x)g(y) + xd(y) = d(x)y + g(x)d(y) and (ii) d(g(x)) = g(d(x)) hold for all x, y ∈ R. The aim of this paper is to explore the commutativity of semiprime rings admitting multiplicative semiderivations.

Semigroup Forum, 1991

In this paper, we give semiring version of some classical results in commutative algebra related to Euclidean rings, PIDs, UFDs, G-domains, and GCD and integrally closed domains.

Discussiones Mathematicae General Algebra and Applications, 2024

Let R be a commutative ring and S a multiplicatively closed subset of R. Hamed and Malek [7] defined an ideal P of R disjoint with S to be an S-prime ideal of R if there exists an s ∈ S such that for all a, b ∈ R if ab ∈ P, then sa ∈ P or sb ∈ P. In this paper, we introduce the notions of S-k-prime and S-k-semiprime ideals of semirings, S-k-m-system, and S-k-p-system. We study some properties and characterizations for S-k-prime and S-k-semiprime ideals of semirings in terms of S-k-m-system and S-kp-system respectively. We also introduce the concepts of S-prime semiring and S-semiprime semiring and study the characterizations for S-k-prime and S-k-semiprime ideals in these two semirings.

Frontiers of Mathematics in China, 2014

A semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. A bounded semiring is a semiring equipped with a compatible bounded partial order. In this paper, properties of zero divisors and prime elements of a bounded semiring are studied. In particular, it is proved that under some mild assumption, the set Z(A) of nonzero zero divisors of A is A \ {0, 1}, and each prime element of A is a maximal element. For a bounded semiring A with Z(A) = A \ {0, 1}, it is proved that A has finitely many maximal elements if ACC holds either for elements of A or for principal annihilating ideals of A. As an application of prime elements, we show that the structure of a bounded semiring A is completely determined by the structure of integral bounded semirings if either |Z(A)| = 1 or |Z(A)| = 2 and Z(A) 2 = 0. Applications to the ideal structure of commutative rings are also considered. In particular, when R has a finite number of ideals, it is shown that the chain complex of the poset I(R) is pure and shellable, where I(R) consists of all ideals of R. Keywords Bounded semiring, zero divisor, prime element, small Z(A), ideal structure of ring MSC 06A11, 13M99