On semifir monoid rings. (English) Zbl 0686.20043

A ring R is said to be a semifir if every finitely generated right ideal is free of unique rank. Let RM be the monoid ring of a non-trivial monoid M over the ring R. If R is a skew field and M is a directed union of free products of free groups and free monoids then it is known that RM is a semifir. W. Dicks has conjectured that the converse is also true. In this paper some necessary conditions on M for RM to be a semifir are given. Furthermore, the author constructs a monoid N (whose unit group is trivial) that satisfies all these conditions but it is not a directed union of free monoids. Therefore, if RN is a semifir, for some skew field R, then RN provides a counter-example to Dicks’ conjecture. Unfortunately, whether or not RN is a semifir for some skew field R is not decided in the present paper.
Reviewer: P.Menal


20M25 Semigroup rings, multiplicative semigroups of rings
16S10 Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)
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