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.

中文翻译：

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半群代数的自注入性

摘要 证明对于一个IC丰裕半群（a 本原丰裕半群；a 本原半单半群）S和域K，若K 0[S] 是右（左）自射，则S是有限正则半群。这扩展和丰富了 Okniński 在正则半群自射代数上的相关结果，肯定地回答了 Okniński 的问题：半群代数 K[S] 是右（分别是左）自射是否意味着 S 是有限的？（半群代数，Marcel Dekker，1990），用于 IC 丰裕半群（原始半单半群；原始丰裕半群）。此外，当 K 0[S] 具有统一性时，我们确定 K 0[S] 的结构是右（左）自射。作为他们的应用，