Commutative Semigroup Rings with Two-Generated Ideals

Abstract

Let R be a commutative ring with identity. If an ideal I of R can be generated by two elements then we say that / is two-generated, and if every ideal of R is twogenerated we say that R has the two-generator property. It is well known that Dedekind domains have the two-generator property and if R has the two-generator property then R has Krull dimension at most 1. In [1] Bass began the study of onedimensional rings which have the two-generator property, and showed that if such a ring R is reduced and has finite integral closure then finitely generated torsion-free Rmodules are isomorphic to direct sums of ideals. One-dimensional reduced rings having the two-generator property and finite integral closure are now often called Bass rings. According to [1, p. 19] one of the original motivations for studying the relationship of the two-generator property to decompositions of /^-modules was to understand the modules over the integral group rings Z[G], where Z is the rational integers, and G a finite abelian group, and it was pointed out there [1, p. 20] that an integral abelian group ring Z[G] has the two-generator property if and only if G has square-free order. In this paper we continue this investigation by determining the onedimensional monoid rings R[S] which have the two-generator property, where S is a commutative cancellative monoid.All rings will be assumed to be commutative, and all monoids S will be commutative and cancellative. By a DVR we mean a discrete rank one valuation domain. The monoids will be written additively, and the elements of R[S] will be written as in [3], that is, a typical element of R[S] will be denoted by r l X** +... + r n X sn , where r t eR and s t eS. We often use the augmentation map R[S] -> R. Recall that this is the i^-algebra homomorphism which maps S to the identity of R. Its kernel is called the augmentation ideal of R[S] and is generated by {X* 1 -X 9 1 g, h e S}, or by {1 -X 9 \ g e S} if S is a group. We refer to [3] for additional properties of monoid rings. The quotient group of a monoid S will be denoted G(S), and if p is a prime integer then the /?-Sylow subgroup of a finitely generated group G will be denoted G p .From the restriction on the dimension we getwhere a denotes the torsion-free rank of G(S) and dim (R) is the Krull dimension of R. Thus there are two cases to consider: dim(i?) = 1, a = 0; and dim(/?) = 0, a = 1.

Numbers of Generators of Ideals in a Group Ring of an Elementary Abelian p-Group

“…(For this see [20,21] and the references listed there.) Similar studies of when a group ring A G satisfies d A G ≤ n for small values of n were given in [2,3,[15][16][17].…”

Artin Rings with Two-Generated Ideals

“…APPLICATION 2. In [8] the commutative one-dimensional monoid rings R[G] which have the two-generator property were determined except in the case that G is a finite abelian group and R is a one-dimensional ring which has some maximal ideals which are also minimal. Since in this case it follows easily that R = R\XR 2 where maximal ideals of R[ have height 1 and R 2 is Artinian, and a product of two rings has the two-generator property if and only if each factor does, it remains to determine the commutative Artinian group rings which have the two-generator property.…”

“…Let (R, M) be a local Artinian ring and let G be a finite abelian group of order n. Then R[G] has the twogenerator property if and only if R has the two-generator property and (i) n is a unit in R if R is not a principal ideal ring, (ii) if R is a principal ideal ring and the characteristic of R/ M is a prime integer p which divides n, then the/?-Sylow subgroup G p of G is = Z/ p l Z and if M 2 ^ 0 then / = 1, unless p -2 where the additional case M = 0 and G 2 = Z/ 2 l Z 0 Z/ 2Z occurs. The proof that if M is a unit of R and R has the two-generator property then R [G] has the two-generator property proceeds by writing R as a homomorphic image of a one-dimensional ring R with the two-generator property as above, and using the characterization in [8] of one-dimensional commutative group rings with the two-generator property.…”

One-Dimensional Monoid Rings with n-Generated Ideals

“…If an ideal I of R can be generated by « elements, then we say that / is n-generated; and, if every ideal of R is «-generated, we say that R has the n-generator property. Determining when a group or monoid ring has the «-generator property has been studied in [1], [3], [4], [6], [7], [8], [9], [11] and [12]. The case n = 1 can be found in [4] or Chapter 19 of [2].…”

“…The case n = 1 can be found in [4] or Chapter 19 of [2]. Monoid (and group) rings with the two-generator property were determined in [8] and [9]. In this note we consider the problem of determining when a one-dimensional monoid ring R [S] has the «-generator property where R is an artinian ring and 5 is a commutative cancellative monoid.…”