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Regular rings and modules

Published online by Cambridge University Press:  09 April 2009

David J. Fieldhouse
David J. Fieldhouse
Affiliation:
University of GuelphGuelph, Ontario, Canada
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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.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 13 , Issue 4 , June 1972 , pp. 477 - 491
Copyright
Copyright © Australian Mathematical Society 1972

References

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