Semigroup algebras that are principal ideal rings. (English) Zbl 0857.16028
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.
Reviewer: M.L.Teply (Milwaukee)
MSC:
16S36 | Ordinary and skew polynomial rings and semigroup rings |
20M35 | Semigroups in automata theory, linguistics, etc. |
20M10 | General structure theory for semigroups |
16D25 | Ideals in associative algebras |
16U30 | Divisibility, noncommutative UFDs |
16P60 | Chain conditions on annihilators and summands: Goldie-type conditions |
16P90 | Growth rate, Gelfand-Kirillov dimension |
16R20 | Semiprime p.i. rings, rings embeddable in matrices over commutative rings |
16U80 | Generalizations of commutativity (associative rings and algebras) |