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Maximal Quotient Rings of Endomorphism Rings of E(RR)-Torsionfree Generators

Published online by Cambridge University Press:  20 November 2018

Tatsuo Izawa
Tatsuo Izawa*
Affiliation:
Shizuoka University, Ohya, Shizuoka, Japan
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Let R be a ring with identity and let H = End (E(RR)) and Q = Dou(E(RR)) = End(HE(RR)). Then Lambek [11] showed that Q is always isomorphic to Qm(R), the maximal right quotient ring of R. And Johnson [10] and Wong-Johnson [26] proved that Qm(R) is regular and right self-injective if and only if R is right non-singular, and then H is isomorphic to Qm(R), too. Moreover, Sandomierski [18] showed that Qm(R) is semi-simple Artinian if and only if R is right finite dimensional and right non-singular. And it is well known that Qm(R) is a quasi-Frobenius ring if and only if E(RR) is a rational extension of RR and the ACC holds on right annihilators of subsets of E(RR).

The purpose of this paper is to give some module-theoretic generalizations of these results. Let PR be an E(PR)-torsionless generator, and let S = End(PR), H = End(E(PR)) and Q = Dou(E(PR)).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 3 , 01 June 1981 , pp. 585 - 605
Copyright
Copyright © Canadian Mathematical Society 1981

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