Regular rings and modules
TL;DR: In this article, it was shown that the ring A is (von Neumann) regular if every left (or right) ideal is pure, which is equivalent to the usual one when A is a PID (= Principal Ideal Domain).
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Abstract: 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.
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Citations
Rugged modules: The opposite of flatness
Engin Büyükaşık,Edgar E. Enochs,J. R. García Rozas,Gizem Kafkas-Demirci,S. R. López-Permouth,Luis Oyonarte +5 more
TL;DR: Relative notions of flatness are introduced as a mean to gauge the extent of the flatness of any given module as mentioned in this paper, and every module is endowed with a flatness domain and, for every ring, the collectio...
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CS-Rickart and dual CS-Rickart objects in abelian categories
TL;DR: In this article, the authors introduce (dual) relative CS-Rickart objects in abelian categories, as common generalizations of the self-CS-RICKART objects and extending (lifting) objects.
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References
Finitistic dimension and a homological generalization of semi-primary rings
TL;DR: In this article, Kaplansky showed that a commutative ring R is left T-nilpotent if, given any sequence {at} of elements in N, there exists an re such that ai • • • an = 0.
On the Free Product of Associative Rings.
TL;DR: Theorem 3.1 of [4] was corrected in this paper, where it was shown that the free product of semifirs R, over a semifir K (under certain conditions) is again a semir.
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Absolutely pure modules
Charles Megibben
- 01 Apr 1970
TL;DR: A module A is shown to be absolutely pure if and only if every finite consistent system of linear equations over A has a solution in A and if A is pure in every injective module containing it as a submodule.
Absolutely pure modules
B. H. Maddox
- 01 Jan 1967
TL;DR: In this article, it was shown that injective and absolutely pure modules are equivalent properties for modules over Dedekind rings and that every module has a maximal maximal injective submodule.