GF-Regular Modules

Abstract

We introduced and studied $\mathrm{GF}$-regular modules as a generalization of $\pi$-regular rings to modules as well as regular modules (in the sense of Fieldhouse). An $R$-module $M$ is called $\mathrm{GF}$-regular if for each $x\in M$ and $r\in R$, there exist $t\in R$ and a positive integer $n$ such that $r^{n}t{r}^{n}x=r^{n}x$. The notion of $G$-pure submodules was introduced to generalize pure submodules and proved that an $R$-module $M$ is $\mathrm{GF}$-regular if and only if every submodule of $M$ is $G$-pure iff $M_{𝔐}$ is a $\mathrm{GF}$-regular $R_{𝔐}$-module for each maximal ideal $𝔐$ of $R$. Many characterizations and properties of $\mathrm{GF}$-regular modules were given. An $R$-module $M$ is $\mathrm{GF}$-regular iff $R/\text{a}\text{n}\text{n}\left(x\right)$ is a $\pi$-regular ring for each $0\ne x\in M$ iff $R/\text{a}\text{n}\text{n}\left(M\right)$ is a $\pi$-regular ring for finitely generated module $M$. If $M$ is a $\mathrm{GF}$-regular module, then $J\left(M\right)=0$.