On weakly injective and divisibility of S-acts

In this paper we investigate actions of a monoid of the form $S=G\dot{\cup} I$, where $G$ is a group and $I$ is an ideal of $S$. So, naturally, every $S$-act can be considered as an $I^1$-act. The central question here is that what is the relation between weakly injective and divisible $I^1$-acts and weakly injective and divisible $S$-acts? We are going to respond this question and show that, given an $S$-act $A$, (principally, finitely generated) weakly injective and divisible property of $A$ is extendable from $I^1$-acts to $S$-acts in general. We also show that if $I$ is strongly regular then an $S$-act $A$ with a unique fixed element $\theta$ is weakly injective if and only if $A$ is injective relative to inclusion $I\hookrightarrow S$. Also if $I^1$ is a left cancellable principal ideal monoid. Then, divisiblity of $A$ as an $I^1$-act implies weakly injectivity of $A$ as an $S$-act