Commutative semigroup rings which are principal ideal rings (original) (raw)
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A commutative ring R is called 2-absorbing (Badawi in Bull. Aust. Math. Soc. 75:417-429, 2007) if for arbitrary elements a, b, c ∈ R, abc = 0 if and only if ab = 0 or bc = 0 or ac = 0. In this paper we study this concept in a more general framework of commutative (multiplicative) semigroups with 0. The results obtained apply to many ring theoretic situations and make it possible to describe similarities and differences among some variants of the notion. We pay a particular attention to graded rings. We also show that a conjecture from (Anderson and Badawi in Commun. Algebra 39:1646-1672, 2011) concerning n-absorbing rings holds for rings with torsion-free additive groups. Keywords Commutative rings • Commutative semigroups • 2-Absorbing rings • Prime ideals 1 Introduction and preliminaries A commutative ring R is called 2-absorbing (cf. [2]) if for arbitrary elements r 1 , r 2 , r 3 of R such that r 1 r 2 r 3 = 0 there are 1 ≤ i = j ≤ 3 for which r i r j = 0. Obviously 2-absorbing rings generalize prime rings. In [2] the structure of such rings was described and it was applied to show that a ring R is 2-absorbing if and only if for Communicated by László Márki.
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Discussiones Mathematicae General Algebra and Applications 31 ( 2011 ) 5 – 23
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Here we introduce the k-bi-ideals in semirings and the intra k-regular semirings. An intra k-regular semiring S is a semiring whose additive reduct is a semilattice and for each a ∈ S there exists x ∈ S such that a + xax = xax. Also it is a semiring in which every k-ideal is semiprime. Our aim in this article is to characterize both the k-regular semirings and intra k-regular semirings using of k-bi-ideals.
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