Type: Article
Publication Date: 1972-06-01
Citations: 8
DOI: https://doi.org/10.1017/s144678870000923x
P. M. Cohn [7] calls a submodule P of the left A -module M pure iff 0 → E ⊗ P → E ⊗ M is exact for all right modules E . This definition of purity, which Cohn [7] has shown to be equivalent to the usual one when A is a PID (= Principal Ideal Domain), was studied in [9] and [10]. Here we show that the ring A is (von Neumann) regular if every left (or right) ideal is pure. This leads us to define regular modules as modules all of whose submodules are pure. The ring A is then regular if all its left (or right) A -modules are regular. A regular socle, analogous to the usual socle is defined. For commutative A , some localization theorems are proved, and used to settle a conjecture of Bass [1] concerning commutative perfect rings.