On algebraically compact semigroup rings
It is known that a group ring R[G] is algebraically compact if and only if G is a finite group and the ring R is algebraically compact [\textit{W. Zimmermann}, J. Pure Appl. Algebra 23, 319-328 (1982; Zbl 0474.16016)]. However, there exist (right) algebraically compact semigroup rings R[S] of infinite semigroups. The authors determine several necessary conditions on the structure of semigroups S of this type, extending the results obtained by the reviewer for the class of right self-injective semigroup rings [Arch. Math. 43, 407-411 (1984; Zbl 0534.20038)]. While these conditions are not sufficient in general, they entirely characterize algebraically compact semigroup rings of completely 0-simple semigroups. We note that it is not known whether S must be finite whenever R[S] is right self-injective. It was proved by the reviewer that, if R is a field, then R[S] is right perfect in this case. This, together with Osofsky's results, implies that S must be finite whenever R[S] is both left and right self-injective. On the other hand, in a forthcoming paper the authors prove that R[S] is left linearly compact (in the discrete topology) if and only if so is R and S is a finite semigroup.
scientific article; zbMATH DE number 4095785