Ideal intrinsic extensions with connections to PI-rings.
type: Item
instance of: scholarly article
title: Ideal intrinsic extensions with connections to PI-rings.
zbMATH Open document ID: 1186.16009
author: Efraim P. Armendariz
author: G. F. Birkenmeier
author: Jae Keol Park
published in: Journal of Pure and Applied Algebra
publication date: 2009-06-04T00:00:00Z
review text: The paper under review studies two quite natural and important generalisations of PI rings (rings satisfying polynomial identities). These are as follows. A ring \(R\) is almost PI if every prime factor of \(R\) is PI. Moreover, \(R\) is intrinsically PI if each nonzero ideal of \(R\) contains a nonzero PI ideal. It is well known that if \(R\) is almost PI then the nonzero ideals of \(R\) have nontrivial intersection with \(\text{Cen}(R)\), the centre of \(R\). This comes to say that almost PI rings are, in a sense, a generalization to the semiprime PI rings. The authors call a ring \(R\) ideal intrinsic over its centre (IIC) if the nonzero ideals intersect \(\text{Cen}(R)\) nontrivially. (Note that sometimes such rings are called rings with large centres.) The paper under review studies, besides almost PI rings, also right bounded rings (that is, rings \(R\) such that each essential right ideal of \(R\) contains an ideal of \(R\) that is essential as a right ideal). First the authors prove that for a semiprime ring \(R\), PI implies almost PI, which in turn implies intrinsically PI. The latter implies also right bounded IIC and it implies IIC. Moreover, the implications are strict. Further on the authors study rings of quotients for intrinsically PI rings. They describe the behaviour of the condition IIC under ring extensions, and also radicals of IIC rings. It is shown that the set of the IIC ideals of a ring is closed under sums and intersections. This of course implies that every semiprime ring possesses a largest IIC ideal. Several additional properties of IIC rings relating them to semiprime ones are deduced as well.
reviewed by: Plamen Koshlukov
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16R20
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16N80
Mathematics Subject Classification ID: 16U70
zbMATH DE Number: 5561728
zbMATH Keywords: semiprime PI rings
zbMATH Keywords: IIC rings
zbMATH Keywords: rings with large centre
zbMATH Keywords: almost PI rings
zbMATH Keywords: polynomial identities
zbMATH Keywords: PI ideals
zbMATH Keywords: ring extensions
MaRDI profile type: Publication
full work available at URL: j.jpaa.2009.01.010
OpenAlex ID: W2033789390
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DOI: 10.1016/J.JPAA.2009.01.010
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description: scientific article; zbMATH DE number 5561728