MONOIDS OVER WHICH ALL REGULAR RIGHT S-ACTS ARE WEAKLY INJECTIVE

Abstract

There have been some study characterizing monoids by homological classication using the properties around projectivity,injectivity, or regularity of acts. In particular Kilp and Knauer([4])have analyzed monoids over which all acts with one of the prop-erties around projectivity or injectivity are regular. However Kilp and Knauer left over problems of characterization of monoids over which all regular right S-acts are (weakly) at, (weakly) injective or faithful. Among these open problems, Liu([3]) proved that all regular right S-acts are (weakly) at if and only if es is a von Neu-mann regular element of S for all s 2 S and e2 = e 2 T, and that all regular right S-acts are faithful if and only if all right ideals eS,e2 = e 2 T, are faithful. But it still remains an open question to characterize over which all regular right S-acts are weakly injective or injective. Hence the purpose of this study is to investigate the relations between regular right S-acts and weakly injective right S-acts, and then characterize the monoid over which all regular right S-acts are weakly injective.