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A note on universally zero-divisor rings

Published online by Cambridge University Press:  17 April 2009

S. Visweswaran
S. Visweswaran
Affiliation:
Department of Mathematics, Saurashtra University Rajkot, India360 005
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Abstract

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In this note we consider commutative rings with identity over which every unitary module is a zero-divisor module. We call such rings Universally Zero-divisor (UZD) rings. We show (1) a Noetherian ring R is a UZD if and only if R is semilocal and the Krull dimension of R is at most one, (2) a Prüfer domain R is a UZD if and only if R has only a finite number of maximal ideals, and (3) if a ring R has Noetherian spectrum and descending chain condition on prime ideals then R is a UZD if and only if Spec (R) is a finite set. The question of ascent and descent of the property of a ring being a UZD with respect to integral extension of rings has also been answered.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 43 , Issue 2 , April 1991 , pp. 233 - 239
Copyright
Copyright © Australian Mathematical Society 1991

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