Semigroup algebras that are principal ideal rings
Let \(K\) be a field and let \(S\) be a semigroup. If the semigroup ring \(K[S]\) is a principal left ideal ring, then \(K[S]\) satisfies a polynomial identity, and hence \(S\) is finitely generated, \(K[S]\) embeds into a matrix ring \(M_n(F)\) over a field extension \(F\) of \(K\), and the Gelfand-Kirillov dimension of \(K[S]\) equals the classical Krull dimension of \(K[S]\). The technical structure of the finite semigroups \(S\) for which \(K[S]\) is a principal ideal ring is completely characterized. If \(K[S]\) is a semiprime principal left ideal ring, then \(K[S]\) is also a principal right ideal ring. Every prime contracted semigroup algebra \(K_0[S]\) must be of the form \(M_n(K)\), \(M_n(K[X])\), or \(M_n(K[X,X^{-1}])\), and the structure of \(S\) is completely determined.
scientific article; zbMATH DE number 923706