‎in this paper some properties of weak regular injectivity for $s$-posets‎, ‎where $s$ is a pomonoid‎, ‎are studied‎. ‎the behaviour‎ ‎of different kinds of weak regular injectivity with products‎, ‎coproducts and direct sums is considered‎. ‎also‎, ‎some‎ ‎characterizations of pomonoids over which all $s$-posets are of‎ ‎some kind of weakly regular injective are obtained‎. ‎further‎, ‎we‎ ‎giv...

we show that every semi-artinian module which is contained in a direct sum of finitely presented modules in $si[m]$, is weakly co-semisimple if and only if it is regular in $si[m]$. as a consequence, we observe that every semi-artinian ring is regular in the sense of von neumann if and only if its simple modules are $fp$-injective.

We show that every semi-artinian module which is contained in a direct sum of finitely presented modules in $si[M]$, is weakly co-semisimple if and only if it is regular in $si[M]$. As a consequence, we observe that every semi-artinian ring is regular in the sense of von Neumann if and only if its simple modules are $FP$-injective.

‎In this paper some properties of weak regular injectivity for $S$-posets‎, ‎where $S$ is a pomonoid‎, ‎are studied‎. ‎The behaviour‎ ‎of different kinds of weak regular injectivity with products‎, ‎coproducts and direct sums is considered‎. ‎Also‎, ‎some‎ ‎characterizations of pomonoids over which all $S$-posets are of‎ ‎some kind of weakly regular injective are obtained‎. ‎Further‎, ‎we‎ ‎giv...

Let S be a semigroup with a left multiplier  on S. A new product on S is defined by  related to S and  such that S and the new semigroup ST have the same underlying set as S. It is shown that if  is injective then where, is the extension of  on  Also, we show that if  is bijective then is amenable if and only if is so. Moreover, if  S completely regular, then is weakly amenable. 

in this article we study two different generalizations of von neumann regularity, namely strong topological regularity and weak regularity, in the banach algebra context. we show that both are hereditary properties and under certain assumptions, weak regularity implies strong topological regularity. then we consider strong topological regularity of certain concrete algebras. moreover we obtain ...

‎‎‎in this paper‎, ‎the‎ notion of injectivity with respect to order dense embeddings in ‎‎the category of $s$-posets‎, ‎posets with a monotone action of a‎ pomonoid $s$ on them‎, ‎is studied‎. ‎we give a criterion‎, ‎like the baer condition for injectivity of modules‎, ‎or skornjakov criterion for injectivity of $s$-sets‎, ‎for the order dense injectivity‎. ‎also‎, ‎we consider such injectivit...