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Pure ring extensions and self FP-injective rings

Published online by Cambridge University Press:  04 October 2011

P. Menal  and
P. Vámos
P. Menal
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Spain
P. Vámos
Affiliation:
Department of Mathematics, University of Exeter, Exeter EX4 4QE
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Extract

In this paper we investigate the following problem: is a ring R right self FP-injective if it has the property that it is pure, as a right R-module, in every ring extension? The answer is ‘almost always’; for example it is ‘yes’ when R is an algebra over a field or its additive group has no torsion. A counter-example is provided to show that the answer is ‘no’ in general. Rings which are pure in all ring extensions were studied by Sabbagh in [4] where it is pointed out that existentially closed rings have this property. Therefore every ring embeds in such a ring. We will show that the weaker notion of being weakly linearly existentially closed is equivalent to self FP-injectivity. As a consequence we obtain that any ring embeds in a self FP-injective ring.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 105 , Issue 3 , May 1989 , pp. 447 - 458
Copyright
Copyright © Cambridge Philosophical Society 1989

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References

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