\(0\)-direct unions of injective \(S\)-systems (Q1377383)

The author considers monoids with zero and right \(S\)-systems (\(S\)-acts) with zero. The categorical concepts are meant in the respective category. The author proves the following Theorem. Let \(E_1\) and \(E_2\) be injective \(S\)-systems. Then \(E_1\sqcup E_2\) is injective iff for any \(0\neq x\in E_1\) and \(0\neq y\in E_2\), there exist \(u\) and \(v\) in \(S\) such that the following conditions hold: (1) \(xu\neq xv\) or \(yu\neq yv\), (2) \(\text{Ann}(xu,xv)\subset\text{Equ}(yu,yv)\), (3) \(\text{Ann}(yu,yv)\subset\text{Equ}(xu,xv)\), (4) \(\text{Ann}(xu,yv)=\text{Ann}(yu,xv)\), where \(\text{Ann}(a,b)=\{s\in S:as=0=bs\}\) and \(\text{Equ}(a,b)=\{t\in S:at=bt\}\) for \(a,b\in A_S\), \(A_S\) a right \(S\)-act, \(E_1\sqcup E_2\) is the coproduct in the category of right \(S\)-acts with zero, i.e. the zero disjoint union of \(E_1\) and \(E_2\).