Hostname: page-component-848d4c4894-r5zm4 Total loading time: 0 Render date: 2024-07-02T20:14:13.824Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Burgess, W. and Menal, P.. Strongly π-regular rings and homomorphisms into them. Comm. Algebra (to appear).Google Scholar
[2] Eklof, P. and Sabbagh, G.. Model-completions and modules. Ann. Math. Logic 7 (1971), 251–295.CrossRefGoogle Scholar
[3] Osofsky, B.. On ring properties of injective hulls. Canad. Math. Bull. 7 (1964), 405–413.CrossRefGoogle Scholar
[4] Sabbagh, G.. Embedding problems for modules and rings with applications to model-companions. J. Algebra 18 (1971), 390–403.CrossRefGoogle Scholar
[5] Stenstrom, B.. Coherent rings and FP-injective modules. J. London Math. Soc. (2) 2 (1970), 232–239.Google Scholar
[6] Warfield, R. B., Jr. Purity and algebraic compactness for modules. Pacific J.Math. 28 (1969), 699–719.CrossRefGoogle Scholar