Abstract
We introduced and studied -regular modules as a generalization of -regular rings to modules as well as regular modules (in the sense of Fieldhouse). An -module is called -regular if for each and , there exist and a positive integer such that . The notion of -pure submodules was introduced to generalize pure submodules and proved that an -module is -regular if and only if every submodule of is -pure iff is a -regular -module for each maximal ideal of . Many characterizations and properties of -regular modules were given. An -module is -regular iff is a -regular ring for each iff is a -regular ring for finitely generated module . If is a -regular module, then .
Highlights
Throughout this paper, unless otherwise stated, R is a commutative ring with nonzero identity and all modules are left unitary
The concept of regular rings was extended firstly to π-regular rings by McCoy [2], recall that a ring R is π-regular if for each r ∈ R, there exist t ∈ R and a positive integer n such that rntrn = rn [2] and secondly to modules in several nonequivalent ways considered by Fieldhouse [3], Ware [4], Zelmanowitz [5], and Ramamurthi and Rangaswamy [6]
It was proved that the following are equivalent for an R-module M: (1) M is GF-regular; (2) every submodule of M is G-pure; (3) R/ann(x) is a π-regular ring for each 0 ≠ x ∈ M; (4) and for each x ∈ M and r ∈ R, there exist t ∈ R and a positive integer n such that rn+1tx = rnx
Summary
Throughout this paper, unless otherwise stated, R is a commutative ring with nonzero identity and all modules are left unitary. It was proved that the following are equivalent for an R-module M: (1) M is GF-regular; (2) every submodule of M is G-pure; (3) R/ann(x) is a π-regular ring for each 0 ≠ x ∈ M; (4) and for each x ∈ M and r ∈ R, there exist t ∈ R and a positive integer n such that rn+1tx = rnx.
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