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.
Highlights
Recall that an algebra R is right self-injective if R is an injective right R-module
R has a left unity. (Left; right) Self-injective algebras are known as the generalizations of Frobenius algebras
It was proved that in some cases, the finiteness of the semigroup is a necessary condition for the semigroup algebra to be right self-injective; for example, the semigroup is an inverse semigroup, a countable semigroup, a regular semigroup, etc
Summary
Recall that an algebra (possibly without unity) R is right self-injective if R is an injective right R-module. It was proved that in some cases, the finiteness of the semigroup is a necessary condition for the semigroup algebra to be right (respectively, left) self-injective; for example, the semigroup is an inverse semigroup, a countable semigroup, a regular semigroup, etc. Guo and Shum [14] proved that the semigroup K[S] of an ample semigroup S is right selfinjective; if and only if K[S] is left self-injective; if and only if K[S] is quasi-Frobenius; if and only if K[S] is Frobenius; if and only if S is a finite inverse semigroup. By inspiring the result of Okniński in [7]: for a regular semigroup S, if K[S] is right (left) self-injective S is finite, we have a natural problem: whether the Okniński problem is valid for IC abundant semigroups? We determine when the algebra of IC abundant semigroups (respectively, primitively semisimple semigroups; primitive abundant semigroups) is semismiple (Theorem 6.2)
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