Arithmetical Semigroup Rings

Abstract

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 S 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].

Noetherian Semigroup Algebras

“…The description of semigroup algebras of cancellative semigroups S that are principal right ideal rings was already shown by Jespers and Wauters in [94]; in case S also is abelian this was done by Gilmer in [54]. The problem of describing principal ideal rings that are commutative semigroup rings R[S] over arbitrary coefficient rings R also has been resolved in [30]; earlier results on this topic can be found in [54,55,62]. …”

Semidistributive modules

“…4 [181,184,190], some applications of distributive rings and modules to formal power series rings were obtained. Distributive group and semigroup rings were studied in [103,42,85,180,182,184,185,187,196]. Distributive quaternion algebras were studied in [197,198,199].…”

Distributive modules and rings and their close analogs

“…In [35,38,44], a number of applications of distributive rings and modules to formal power series rings were obtained. Distributive group and semigroup rings were studied i n [14,111,95,34,36,38,39,41,51]. Distributive quaternion algebras were studied in [52][53][54].…”