Notes on pure projective modules

关于纯投射模的注记(英文)

Let R be an associated ring with identity. A new equivalent characterization of pure projective left R-modules is given by applying homological methods. It is proved that a left R-module P is pure projective if and only if for any pure epimorphism E→M→0, where E is pure injective, HomR（P, E）→HomR（P, M）→0 is exact. Also, we obtain a dual result of pure injective left R-modules. Furthermore, it is shown that every pure projective left R-module is closed under pure submodule if and only if every pure injective left R-module is closed under pure epimorphic image.

R表示有单位元的结合环.通过同调的方法,给出了纯投射左R模的一个新的等价刻画.证明了左R模P是纯投射的当且仅当对任意纯满射E→M→0,其中E是纯内射的,HomR(P,E)→HomR(P,M)→0是正合的.同时,关于纯内射模的对偶结果也是成立的.最后,作为应用,证明了每一纯投射左R模在纯子模下封闭当且仅当每一纯内射左R模在纯满像下封闭.