The isomorphism problem for commutative monoid rings
The isomorphism problem for monoid rings asks whether two monoids are isomorphic if they have isomorphic monoid rings. Here all monoids are commutative, cancellative, and torsionfree and all rings are commutative with 1. The main result of this paper is that if \(M\) and \(N\) are finitely generated monoids and \(R\) is a ring with \(R[M]\approx R[N]\) as \(R\)-algebras, then \(M\approx N\) if either (1) \(M\) is normal, (2) \(M\) and \(N\) have only trivial units and \(R[M]\approx R[N]\) as augmented \(R\)-algebras, or (3) \(M\) is homogeneous. The proof is accomplished by extending and correcting arguments in a paper of \textit{A. S. Demushkin} [Vestn. Mosk. Univ., Ser. I 1982, No. 2, 80-87 (1982; Zbl 0507.14033)]. As an application, the author also gives a positive answer to the isomorphism problem for discrete Hodge algebras.
scientific article; zbMATH DE number 1308187
scientific article