A ring which is a sum of two PI subrings is always a PI ring
type: Item
instance of: scholarly article
title: A ring which is a sum of two PI subrings is always a PI ring
zbMATH Open document ID: 1396.16015
author: Marek Kępczyk
published in: Israel Journal of Mathematics
publication date: 2017-11-17T00:00:00Z
review text: In 1962 the following problem was posed by \textit{O. H. Kegel} [Math. Ann. 149, 258--260 (1963; Zbl 0106.25402)]: If a ring \(R\) is a sum of two subrings \(R_1\) and \(R_2\) and both \(R_1\) and \(R_2\) satisfy polynomial identities, is then \(R\) a PI ring? (In fact Kegel originally posed it in a somewhat different form but the essence is the same.) This problem attracted the attention of various algebraists, and there have been several partial results. The paper under review settles the problem in its generality. Namely, the author proves that if \(R\) is an associative ring (not necessarily with 1), and \(R=R_1+R_2\) is a sum of two subrings, then \(R\) is PI whenever both \(R_1\) and \(R_2\) are PI. In order to obtain this important result the author makes use of a theorem proved in [the author and \textit{E. R. Puczyłowski}, Commun. Algebra 29, No. 5, 2059--2065 (2001; Zbl 0993.16017)]. The theorem is the following. Let \(T\) be a class of rings that is closed under direct powers and under homomorphisms. If each nontrivial prime ring in \(T\) contains a nontrivial one-sided ideal which is PI as a ring then \(T\) consists of PI rings. It is worth mentioning two interesting corollaries to the main theorem of the paper. {\parindent=6mm \begin{itemize}\item[1.] Let \(R\) be a sum of two nil and PI subrings \(R_1\) and \(R_2\). Then \(R\) is also nil (and PI). \item[2.] Let \(R=R_1+R_2+R_3\) be a ring that is a sum of three PI subrings. If at least one of these subrings is one-sided ideal in \(R\) then \(R\) is PI. \end{itemize}}
reviewed by: Plamen Koshlukov
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16R50
zbMATH DE Number: 6808580
zbMATH Keywords: PI rings
zbMATH Keywords: nil subrings
MaRDI profile type: Publication
full work available at URL: s11856-017-1554-3
OpenAlex ID: W2737674362
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DOI: 10.1007/S11856-017-1554-3
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description: scientific article; zbMATH DE number 6808580