In "Semigroup Algebras," Okniński posed the following question: characterize semigroup algebras that are hereditary. In this paper we describe the (prime contracted) semigroup algebras K[S] that are hereditary and Noetherian when S is either a Malcev nilpotent monoid, a cancellative monoid or a monoid extension of a finite non-null Rees matrix semigroup. Furthermore, for the class of monoids which have an ideal series with factors that are non-null Rees matrix semigroups, we obtain an upper bound for the global dimension of its contracted semigroup algebra.

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Journal Journal of Algebra
Jespers, E. (Eric), & Wang, Q. (2000). Hereditary semigroup algebras. Journal of Algebra, 229(2), 532–546.