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Arithmetical Semigroup Rings

Published online by Cambridge University Press:  20 November 2018

Bonnie R. Hardy  and
Thomas S. Shores
Bonnie R. Hardy
Affiliation:
University of Nebraska, Lincoln, Nebraska
Thomas S. Shores
Affiliation:
University of Nebraska, Lincoln, Nebraska
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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].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 6 , 01 December 1980 , pp. 1361 - 1371
Copyright
Copyright © Canadian Mathematical Society 1980

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