On injective envelopes of semimodules over semirings (Q2816884)

Let \((S,+,\cdot)\) be an additively commutative semiring with absorbing zero \(0\) and identity \(1\). Then \(S\) is called \textit{semizeroic} if for every \(a\in S\) there are \(x,y\in S\) such that \(a + x + y = y\). The first main result of the paper states that the following conditions are equivalent: {\parindent=0.6cm \begin{itemize}\item[(1)] Each simple right (left) \(S\)-semimodule has an injective envelope, i.e., can be embedded in an injective \(S\)-semimodule. \item[(2)] Each additively idempotent simple right (left) \(S\)-semimodule has an injective envelope. \item[(3)] Each simple right (left) \(S\)-module has an injective envelope. \item[(4)] Each right (left) \(S\)-module has an injective envelope. \item[(5)] \(S\) is semizeroic. NEWLINENEWLINE\end{itemize}}Moreover, \(S\) is called \textit{additively \(\pi\)-regular} if for any \(a\in S\) there exists \(x\in S\) such that \(na+x+na = na\) for some \(n\geq 1\). Then the second main result states that the following conditions are equivalent: {\parindent=0.6cm \begin{itemize}\item[(1)] Every additively idempotent right (left) \(S\)-semimodule has an injective envelope. \item[(2)] Every additively regular right (left) \(S\)-semimodule has an injective envelope. \item[(3)] \(S\) is additively \(\pi\)-regular. NEWLINENEWLINE\end{itemize}}