Affiliation:
1. College of Science and Technology, Jiangxi Normal University , Nanchang , 330022 , China
2. College of Mathematics and Information Science, Jiangxi Normal University , Nanchang , 330022 , China
Abstract
Abstract
It is proved that for an IC abundant semigroup (a primitive abundant semigroup; a primitively semisimple semigroup) S and a field K, if K
0[S] is right (left) self-injective, then S is a finite regular semigroup. This extends and enriches the related results of Okniński on self-injective algebras of regular semigroups, and affirmatively answers Okniński’s problem: does that a semigroup algebra K[S] is a right (respectively, left) self-injective imply that S is finite? (Semigroup Algebras, Marcel Dekker, 1990), for IC abundant semigroups (primitively semisimple semigroups; primitive abundant semigroups). Moreover, we determine the structure of K
0[S] being right (left) self-injective when K
0[S] has a unity. As their applications, we determine some sufficient and necessary conditions for the algebra of an IC abundant semigroup (a primitively semisimple semigroup; a primitive abundant semigroup) over a field to be semisimple.
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