Semiprimeness of semigroup algebras

Abundant semigroups originate from p.p. rings and are generalizations of regular semigroups. The main aim of this paper is to study the primeness and the primitivity of abundant semigroup algebras. We introduce and study D ∗ {{\mathcal{D}}}^{\ast } -graphs and Fountain matrices of a semigroup. Based on D ∗ {{\mathcal{D}}}^{\ast } -graphs and Fountain matrices, we determine when a contracted semigroup algebra of a primitive abundant semigroup is prime (respectively, semiprime, semiprimitive, or primitive). It is well known that for any algebra A {\mathcal{A}} with unity, A {\mathcal{A}} is primitive (prime) if and only if so is M n ( A ) {M}_{n}\left({\mathcal{A}}) . Our results can be viewed as some kind of generalizations of such a known result. In addition, it is proved that any contracted semigroup algebra of a locally ample semigroup whose set of idempotents is locally finite (respectively, locally pseudofinite and satisfying the regularity condition) is isomorphic to some contracted semigroup algebra of primitive abundant semigroups. Moreover, we obtain sufficient and necessary conditions for these classes of contracted semigroup algebras to be prime (respectively, semiprime, semiprimitive, or primitive). Finally, the structure of simple contracted semigroup algebras of idempotent-connected abundant semigroups is established. Our results enrich and extend the related results on regular semigroup algebras.