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Non-local rings whose ideals are all quasi-injective

Published online by Cambridge University Press:  17 April 2009

G. Ivanov
G. Ivanov
Affiliation:
Department of Pure Mathematics, School of General Studies, Australian National University, Canberra, ACT.
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Abstract

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A ring is a left Q-ring if all of its left ideals are quasi-injective. For an integer m ≤ 2, a sfield D, and a null D-algebra V whose left and right D-dimensions are both equal to one, let H(m, D, V) be the ring of all m x m matrices whose only non-zero entries are arbitrary elements of D along the diagonal and arbitrary elements of V at the places (2, 1), …, (m, m-l) and (l, m). We show that the only indecomposable non-local left Q-rings are the simple artinian rings and the rings H(m, D, V). An arbitrary left Q-ring is the direct sum of a finite number of indecomposable non-local left Q-rings and a Q-ring whose idempotents are all central.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 6 , Issue 1 , February 1972 , pp. 45 - 52
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Faith, Carl, Lectures on injective modules and quotient rings (Lecture Notes in Mathematics, 49. Springer-Verlag, Berlin, Heidelberg, New York, 1967).CrossRefGoogle Scholar
[2]Jain, S.K., Mohamed, S.H. and Singh, Surjeet, “Rings in which every right ideal is quasi-injactive”, Pacific J. Math. 31 (1969), 73–79.CrossRefGoogle Scholar