Derivations of prime rings
type: Item
instance of: scholarly article
title: Derivations of prime rings
zbMATH Open document ID: 0537.16030
DOI: 10.1007/BF01979650
author: A. Z. Popov
published in: Algebra and Logic
publication date: 1983-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/187208
review text: This paper has two types of results. First, there are proved the following theorems about rings of constants of finite dimensional \(\partial\)-Lie algebras of external derivations of rings with positive characteristic: Theorem 1. Let L be a finite-dimensional \(\partial\)-Lie algebra of external derivations of the prime ring R with positive characteristic. Then the ring of constants \(R^ L\) is a left Goldie ring if and only if R is a left Goldie ring. Theorem 2. Let R be a prime ring. Then \(Q_ M(R)^ L=Q_ M(R^ L)\), where \(Q_ M(B)\) denotes the two-sided Martindale quotient ring of B and R,L are as in theorem 1. Similar results for rings of invariants of finite groups were obtained by \textit{V. K. Kharchenko} [Algebra Logika 16, 313-363 (1977; Zbl 0397.16037)]. Theorem 1 answers Question 2 from \textit{V. K. Kharchenko} [Izv. Akad. Nauk SSSR, Ser. Mat. 45, 435-462 (1981; Zbl 0469.16019)]. Secondly the author gives complete answers to the following questions which were formulated by \textit{I. N. Herstein} [(1): Can. Math. Bull. 21, 369-370 (1978; Zbl 0412.16018)]: (2) J. Algebra 60, 567-574 (1979; Zbl 0436.16014)]: 1) Let R be a prime ring and R satisfies an identity \(S_ n(d(x_ 1),...,d(x_ n))=0,\) where d is a non zero derivation on R. Has R the identity \(S_ n(x_ 1,...,x_ n)=0?\) 2) Let R be a prime ring, d be a derivation on R such that \((d(x))^ n=0\) for an integer \(n\geq 1\). Holds \(d=0?\) Question 1 has answer ''yes'', if \(n=2\) [\textit{I. N. Herstein}, (1)], and ''no'', if \(n\geq 9\) [\textit{A. Kovacs}, Can. Math. Bull. 22, 339-344 (1979; Zbl 0436.16015)]. Question 2 has answer ''yes'', if d is an inner derivation. In the paper under review it is proved that the ring of infinite row- finite matrices satisfies the identity \(S_ 3(d(x_ 1),d(x_ 2),d(x_ 3))\), where d is an inner derivation, defined by the matrix unit \(e_{12}\), yet it is well-known, that this ring not satisfies any identities. Furthermore, there is proved Proposition 3. Let R be a p…
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16Rxx
Mathematics Subject Classification ID: 16N60
zbMATH DE Number: 3853315
zbMATH Keywords: rings of constants
zbMATH Keywords: external derivations
zbMATH Keywords: finite-dimensional \(\partial\)- Lie algebra
zbMATH Keywords: left Goldie ring
zbMATH Keywords: Martindale quotient ring
zbMATH Keywords: inner derivation
zbMATH Keywords: identities
zbMATH Keywords: prime ring
zbMATH Keywords: differential identity
MaRDI profile type: Publication
cites work: Q3922817
cites work: Q4180503
cites work: Q3874360
cites work: Differential identities of semiprime rings
cites work: Q4186484
cites work: A Note on Derivations
cites work: Center-like elements in prime rings
cites work: Q5639810
cites work: Prime rings satisfying a generalized polynomial identity
cites work: On Derivations in Prime Rings and a Question of Herstein
OpenAlex ID: W2061341861
description: scientific article; zbMATH DE number 3853315