On ℵ₀-Injectivity

Abstract

In this note we investigate ℵ₀-injectivity of rings and modules and review the literature around this topic. We observe that several characterizations of rings by injectivity can be expressed by ℵ₀-injectivity. Moreover we point out that the following three classes of rings are not axiomatizable: the right ℵ₀-self-injective rings, the right ℵ₀-self-injective regular rings and the regular Baer rings.

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Acknowledgments

This paper has been written while the author was a visiting student at Heinrich Heine University in Düsseldorf. He would like to thank Professor R. Wisbauer for his warm hospitality and the very helpful discussions on the subject and Professor M. Prest for his interest in my work. The author would also like to thank Professor O. A. S. Karamzadeh, his supervisor who suggested the topic of this article, and Dr. M. Motamedi, his advisor, for their suggestions and encouragements. Moreover, the author is grateful to the referee for careful reading.

Notes

^#Communicated by R. Wisbauer.