Zhanmin Zhu: Rings related to 𝒮-projective modules, 439-449

Abstract:

Let $R$ be a ring and $\mathcal {S}$ be a class of some finitely generated left $R$-modules. Then a left $R$-module $M$ is called $\mathcal {S}$-projective if for every homomorphism $f : S \rightarrow M$, where $S\in \mathcal {S}$ , there exist homomorphisms $g : S\rightarrow F$ and $h : F\rightarrow M$ such that $f = hg$, where $F$ is a free module . In this paper, we give some characterizations of the following four classes of rings: (1). every injective left $R$-module is $\mathcal {S}$-projective; (2). every left $R$-module has a monic $\mathcal {S}$-projective preenvelope; (3). every submodule of a projective left $R$-module is $\mathcal {S}$-projective; (4). every left $R$-module has an epic $\mathcal {S}$-projective envelope. Some applications are given.

Key Words: $\mathcal {S}$-projective modules, $\mathcal {S}$-$\Pi$-coherent rings, $\mathcal {S}$-F rings, $\mathcal {S}$-semihereditary rings.

2010 Mathematics Subject Classification: Primary 16D40; Secondary 16P70.