The minimal prime spectrum of rings with annihilator conditions.
type: Item
instance of: scholarly article
title: The minimal prime spectrum of rings with annihilator conditions.
zbMATH Open document ID: 1218.16001
author: Chan Yong Hong
author: Nam Kyun Kim
author: Yang Lee
author: Pace P. Nielsen
published in: Journal of Pure and Applied Algebra
publication date: 2009-04-16T00:00:00Z
review text: A ring \(R\) is said to have the right annihilator condition (a.c.) if the right annihilator of any finitely generated ideal of \(R\) equals the right annihilator of some principal ideal. Many rings are shown to have the (right and left) (a.c.), such as semiprime rings with finitely many minimal prime ideals; biregular rings; skew polynomial rings (with an automorphism) over semiprime rings. Further, (a.c.) is preserved in matrix rings and in classical quotient rings. The minimal prime spectrum of \(R\) is the set \(\text{Min}(R)\) of minimal prime ideals of \(R\), equipped with the relative hull-kernel (Zariski) topology from \(\text{Spec}(R)\). The authors prove that if \(R\) is a reduced ring with classical right quotient ring \(Q(R)\), the following conditions are equivalent: (1) \(Q(R)\) is strongly regular; (2) \(\text{Min}(R)\) is compact and \(R\) has (a.c.); (3) \(\text{Min}(R)\) is compact and every finitely generated ideal of \(R\) consisting of left (right) zero-divisors has nonzero right (left) annihilator. This generalizes results in the commutative case due to \textit{M. Henriksen} and \textit{M. Jerison} [Trans. Am. Math. Soc. 115, 110-130 (1965; Zbl 0147.29105)], \textit{Y. Quentel} [Bull. Soc. Math. Fr. 99, 265-272 (1971); erratum ibid. 100, 461 (1972; Zbl 0215.36803)], and \textit{J. A. Huckaba} and \textit{J. M. Keller} [Pac. J. Math. 83, 375-379 (1979; Zbl 0388.13001)].
reviewed by: Kenneth R. Goodearl
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16W80
zbMATH DE Number: 5544397
zbMATH Keywords: right annihilator condition
zbMATH Keywords: right annihilators
zbMATH Keywords: finitely generated ideals
zbMATH Keywords: minimal prime ideals
zbMATH Keywords: semiprime rings
zbMATH Keywords: minimal prime spectra
zbMATH Keywords: reduced rings
MaRDI profile type: Publication
full work available at URL: j.jpaa.2009.01.005
OpenAlex ID: W1964421873
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DOI: 10.1016/J.JPAA.2009.01.005
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description: scientific article; zbMATH DE number 5544397