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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()
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() 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.
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