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

Abstract

There have been some study characterizing monoids by
homological classification 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) flat, (weakly) injective
or faithful. Among these open problems, Liu([3]) proved that all
regular right S-acts are (weakly) flat if and only if es is a von Neu-mann regular element of S for all $s \in S$ and $e^2 = e \in T$ , and that all regular right S-acts are faithful if and only if all right ideals $eS$, $e^2 = e \in T$ , 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.