Monoids characterized by their quasi-injective S-systems (Q579433)

Let S be a monoid. A right S-system \(M_ S\) is called finitely injective if for every S-monomorphism \(f: X_ S\to Y_ S\), where X is a finitely generated S-system, and for every S-homomorphism \(g: X_ S\to M_ S\), there exists an S-homomorphism \(h: Y_ S\to M_ S\) such that \(hf=g\). \(M_ S\) is quasi-injective if for \(N_ S\subseteq M_ S\) and every S- homomorphism \(f: N_ S\to M_ S\), there exists an S-homomorphism \(g: M_ S\to M_ S\) such that \(g|_ N=f\). M is countably \(\Sigma\)- injective if the direct sum of countably many copies of M is injective. S is right noetherian if and only if S satisfies the ascending chain condition for right ideals. For a monoid S, the following conditions are shown to be equivalent in the category of centered S-systems: (1) Each direct sum of injective S-systems is injective. (2) Each direct sum of weakly injective S-systems is weakly injective. (3) Each injective S- system is countably \(\Sigma\)-injective. (4) Each finitely injective S- system is weakly injective. (5) S is noetherian. Furthermore, it has been proved that each direct sum of quasi-injective S-systems is quasi- injective if and only if S is noetherian and each quasi-injective S- system is injective.