On radicals of rings which are sums of two subrings
type: Item
instance of: scholarly article
title: On radicals of rings which are sums of two subrings
zbMATH Open document ID: 0860.16016
DOI: 10.1007/BF01323977
published in: Archiv der Mathematik
publication date: 1997-02-18T00:00:00Z
review text: Rings which are sums of two subrings have been studied in many papers. In this paper, the authors study the radicals of such rings. They obtain new results concerning the open questions of whether a ring which is a sum of two prime radical subrings must be prime radical. The authors use the symbol ``\(A\in S\)'' or ``\(A\) is an \(S\)-ring'' to mean a ring \(A\) has property \(S\). A radical \(S\) is hereditary if whenever a ring \(A\in S\) and \(I\) is an ideal of \(A\), then \(I \in S\). Hereditary radicals containing the prime radical \(\beta\) are called supernilpotent. A radical \(S\) is right strong if for every ring \(A\), \(S(A)\) contains all right \(S\)-ideals of \(A\). Given a ring \(A\), denote the left-annihilator of \(A\) by \(l(A)\). Throughout this paper \(R\) is a ring and \(R_1\), \(R_2\) are subrings of \(R\) such that \(R=R_1+R_2\). The authors prove Theorem 1: If \(R_1\in\beta\) and \(R_2\in S\), where \(S\) is a right strong and supernilpotent radical, then \(l(R_1)\subset S(R)\). They give an example to show the assumption that \(R_1\in\beta\) cannot be omitted from this theorem. It is known that if a radical \(S\) containing \(\beta\) is left or right strong and left or right hereditary, then it is both left and right strong as well as left and right hereditary. Radicals of that type are called \(N\)-radicals. The problem whether the nil radical is an \(N\)-radical is still open; it is called Koethe's problem. The authors' Theorem 1 enables them to prove if \(R_1\) is a nilpotent ring and \(S\) is an \(N\)-radical then \(R\in S\). A ring \(A\) is left \(T\)-nilpotent if for every non-zero homomorphic image \(A'\) of \(A\), \(l(A')\neq 0\). The authors prove that if \(R_1\) is a nil PI ring and \(R_2\in S\), where \(S\) is an \(N\)-radical, then \(R\in S\). This theorem enables them to prove the following corollaries: (i) Let \(S\) be an \(N\)-radical. If \(R_1\) is a nil ring of bounded index and \(R_2\in S\), then \(R\in S\). (ii) If \(R_1\)…
Mathematics Subject Classification ID: 16N80
Mathematics Subject Classification ID: 16N40
zbMATH DE Number: 937064
zbMATH Keywords: hereditary radicals
zbMATH Keywords: sums of two subrings
zbMATH Keywords: prime radical subrings
zbMATH Keywords: supernilpotent radicals
zbMATH Keywords: nil radical
zbMATH Keywords: \(N\)-radicals
zbMATH Keywords: Koethe's problem
zbMATH Keywords: nilpotent rings
zbMATH Keywords: nil PI rings
zbMATH Keywords: nil rings of bounded index
zbMATH Keywords: commutative nil rings
zbMATH Keywords: right \(T\)-nilpotent rings
zbMATH Keywords: Wedderburn radical rings
author: Marek Kępczyk
author: E. R. Puczyłowski
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01323977
OpenAlex ID: W2074042189
description: scientific article; zbMATH DE number 937064