Hostname: page-component-cb9f654ff-pvkqz Total loading time: 0 Render date: 2025-08-31T09:03:49.647Z Has data issue: false hasContentIssue false
  • English
  • Français

Pure-semisimplicity is preserved under elementary equivalence

Published online by Cambridge University Press:  18 May 2009

Maher Zayed
Maher Zayed
Affiliation:
Department of Mathematics, University of Bahrain, P.O. Box 32038, Isa Town, State of Bahrain
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In the present note, Σr denotes the class of all right pure semisimple rings (= right pure global dimension zero). It is known that if R ∈ Σr, then R is right artinian and every indecomposable right R-module is finitely generated. The class Σr is not closed under ultraproducts [4]. While Σr is closed under elementary descent (i.e. if S ∈ Σr and R is an elementary subring of S then R ∈ σr) [4], it is an open question whether right pure-semisimplicity is preserved under the passage to ultrapowers [4, Prob. 11.16]. In this note, this question is answered in the affirmative.

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 36 , Issue 3 , September 1994 , pp. 345 - 346
Copyright
Copyright © Glasgow Mathematical Journal Trust 1994

References

REFERENCES

1.Bell, J. L. and Slomson, A. B., Models and Ultraproducts. (Amsterdam: North-Holland, 1974).Google Scholar
2.Fuller, K. R., On rings whose left modules are direct sums of finitely generated modules. Proc. Amer. Math. Soc. 54 (1976), 39–44.CrossRefGoogle Scholar
3.Herrmann, C., Jensen, C. U. and Lenzing, H., Applications of model theory to representations of finite-dimensional algebras. Math. Z. 178 (1981), 83–98.CrossRefGoogle Scholar
4.Jensen, C. U. and Lenzing, H., Model theoretic algebra with particular emphasis on fields, rings, modules and finite dimensional algebras. (Gordon and Breach Science Pub. New York, 1989).Google Scholar
5.Prest, M., Duality and pure-semisimple rings. J. London Math. Soc. 38 (1988), 403–409.CrossRefGoogle Scholar
6.Simson, D., Fanctor categories in which every flat object is projective. Bull. Acad. Polon. 22 (1974), 375–380.Google Scholar
7.Zimmermann-Huisgen, B. and Zimmermann, W.: On the sparsity of representations of rings of pure global dimension zero. Trans. Amer. Math. Soc, 320 (1990), 695–711.CrossRefGoogle Scholar