Structure of prime finitely presented monomial algebras

AbstractThe structure of a finitely presented monomial algebra K[X]/K[I] over a field K is described. Here X is a finitely generated free monoid and I is a prime ideal of X that is finitely generated. As an application, a new structural proof of the recent result of Bell and Pekcagliyan [J. Bell, P. Pekcagliyan, Primitivity of finitely presented monomial algebras, preprint, arXiv: 0712.0815v1] on the primitivity of such algebras is presented, which yields a positive solution to the trichotomy problem, raised by Bell and Smoktunowicz [J. Bell, A. Smoktunowicz, The prime spectrum of algebras of quadratic growth, J. Algebra 319 (2008) 414–431], in the finitely presented case. Our approach is based on a new result on the form of prime Rees factors of semigroups satisfying the ascending chain condition on one-sided annihilators and on its refinement in the case of finitely presented factors of the form X/I

Faithful linear representations of bands

A band is a semigroup consisting of idempotents. It is proved that for any field K and any band S with finitely many components, the semigroup algebra K [S] can be embedded in upper triangular matrices over a commutative K-algebra. The proof of a theorem of Malcev [4, Theorem 10] on embeddability of algebras into matrix algebras over a field is corrected and it is proved that if S = F ∪ E is a band with two components E, F such that F is an ideal of S and E is finite, then S is a linear semigroup. Certain sufficient conditions for linearity of a band S, expressed in terms of annihilators associated to S, are also obtained