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Commutative Self-Injective Rings

Published online by Cambridge University Press:  20 November 2018

Surjeet Singh  and
Kamlesh Wasan
Surjeet Singh
Affiliation:
Aligarh Muslim University, Aligarh (U. P.), India
Kamlesh Wasan
Affiliation:
Aligarh Muslim University, Aligarh (U. P.), India Mata Sundri College for Women, New Delhi, India
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All rings considered here are commutative containing at least two elements, but may not have identity. A ring R is said to be selfinjective if R as an R-module is injective. A ring R is said to be pre-selfinjective if every proper homomorphic image of R is self-injective [9]. Study of pre-self-injective rings was initiated by Levy [10], who established a characterization of Noetherian pre-self-injective rings with identity in terms of other well-known types of rings. Recently Klatt and Levy [9] have characterized all pre-self-injective rings with identity. In this paper we are mainly interested in Noetherian rings. For the sake of convenience we shall call a pre-self-injective ring an (I)-ring. A ring R will be said to be a (PMI)-ring if for each proper prime ideal P with P2 ≠ 0, the ring R/P2 is self-injective. Clearly, an (I)-ring is a (PMI)-ring.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 6 , 01 December 1970 , pp. 1101 - 1108
Copyright
Copyright © Canadian Mathematical Society 1970

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