Some characterizations of regular modules
Two characterizations of regularity in the sense of \textit{J. Zelmanowitz} [Trans. Am. Math. Soc. 163, 341-355 (1972; Zbl 0227.16022)] are proved. (A left R-module M is Zelmanowitz-regular provided that for each \(x\in M\) there exists a homomorphism \(h_ x: M\to R\) such that \(h_ x(x)x=x.)\) First, a module M is Zelmanowitz-regular if and only if every homomorphism f: \(L\to M\) is locally split in the sense that for each \(x\in f(L)\), there exists a homomorphism \(g_ x: M\to L\) such that \(fg_ x(x)=x\). Second, M is Zelmanowitz-regular if and only if (a) every cyclic submodule of M is a direct summand, and (b) M is locally projective in the sense that every epimorphism onto M is locally split.
scientific article; zbMATH DE number 4189942