Abundant semigroup algebras which are Azumaya

It is proved that any Azumaya semigroup algebra of an abundant semigroup over a field is a subdirect product of matrix algebras over Azumaya semigroup algebras of cancellative monoids. In particular, it is proved that for an abundant semigroup S with finitely many idempotents and a field K , K_0[S] is Azumaya if and only if K_0[S] is isomorphic to the direct sum of finite matrix algebras of Azumaya algebras of cancellative submonoids. This gives an affirmative answer to an open problem of Okniński in his monograph ( Semigroup Algebra , Marcel Dekker, New York, 1991) for the case that the semigroup is a finite abundant semigroup.