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.

证明了域上任何丰裕半群的Azumaya半群代数都是抵消幺半群的Azumaya半群代数上矩阵代数的子直积。特别地，证明了对于具有有限多个幂等项和域K的丰富半群S，\(K_0[S]\)是 Azumaya 当且仅当\(K_0[S]\)同构于消元子群的 Azumaya 代数的有限矩阵代数。这对 Okniński 在他的专着 (Semigroup Algebra , Marcel Dekker, New York, 1991) 中的一个开放问题给出了肯定的答案，因为该半群是一个有限充裕半群。