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April 2020 On the socle of a commutative ring and Zariski topology
Ali Taherifar
Rocky Mountain J. Math. 50(2): 707-717 (April 2020). DOI: 10.1216/rmj.2020.50.707

Abstract

This paper concerns the coincidence of the socle of a semiprimitive ring (or just a reduced one, in some cases) with the intersection of all essential prime, essential minimal prime, or essential maximal ideals of the ring. More precisely, we prove first that the socle of a reduced ring coincides with the intersection of all essential minimal prime ideals if and only if every minimal prime (or, equivalently, every prime) ideal is either essential or it is a direct summand which is also a maximal ideal. Next, we show that the socle of a semiprimitive ring R is equal to the intersection of all essential maximal ideals of R (i.e., Soc max ( R ) ) if and only if the set of isolated points of Max ( R ) with the Zariski topology contains no infinite basic open set. Whenever R is a semiprimitive c.a.c. ring, we prove that for every essential ideal I of R containing Soc max ( R ) , I Soc max ( R ) is essential in R Soc max ( R ) if and only if the set of isolated points of Max ( R ) is finite. We apply this result to rings of continuous real-valued functions on a topological space.

Citation

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Ali Taherifar. "On the socle of a commutative ring and Zariski topology." Rocky Mountain J. Math. 50 (2) 707 - 717, April 2020. https://doi.org/10.1216/rmj.2020.50.707

Information

Received: 16 February 2018; Revised: 25 February 2019; Accepted: 19 July 2019; Published: April 2020
First available in Project Euclid: 29 May 2020

zbMATH: 07210991
MathSciNet: MR4104406
Digital Object Identifier: 10.1216/rmj.2020.50.707

Subjects:
Primary: 13A15
Secondary: 54C40

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium

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Vol.50 • No. 2 • April 2020
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