Power values of generalized skew derivations with annihilator conditions on Lie ideals
type: Item
instance of: scholarly article
title: Power values of generalized skew derivations with annihilator conditions on Lie ideals
zbMATH Open document ID: 1460.16021
author: Münevver Pınar Eroğlu
author: N. Argaç
published in: Bulletin of the Iranian Mathematical Society
publication date: 2021-01-05T00:00:00Z
review text: Let \(R\) be a ring and let \(M\) be a simple right \(R\)-module. Now let \(m\) be a non-zero member of \(M\) such that \(mR=M\). Hence, there exists a non-zero element \(n\) of \(M\) such that there is an element of \(R\) that induces and endomorphism of \(M\) transforming \(m\) to \(n\). This condition motivates the natural question which asked whether this can be generalized to arbitrary finite-tuples of element. Furthermore, what are the necessary and sufficient conditions on the tuples \((x_1, x_2,\dots, x_n)\) and \((y_1, y_2,\dots, y_n)\) separately, so there exists an element of \(R\) with the property that \(x_ir=y_i\) for all \(i \in \{1,2,\dots,n\}\) where \(r \in R\). A well-known theorem which is called the Jacobson density theorem gives a positive answer on tuples affirmatively which provided that the \(x_i\) are linearly independent over \(D\), where \(D\) is a division ring. In this paper, the authors use the extended Jacobson density theorem to characterize the forms of generalized skew derivations \(\delta\) and \(\varrho\) of a prime ring \(R\) with maximal right ring of questions \(Q\), extended centroid \(C\), and non-commutative Lie ideal \(L\) in case of the following equation \(p(\delta(u)^{l_1}\varrho{u}^{l_2}\delta{u}^{l_3}\varrho{u}^{l_4}\dots\varrho(u)^{l_k})^n=0,\) for all \(u \in L,\) where \(0\neq p \in R, l_1, l_2,\dots,l_k\) are fixed nonnegative integers with \(l_1 \neq 0\) and \(n\) is a positive fixed integer.
reviewed by: Puguh Wahyu Prasetyo
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16R20
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16W55
zbMATH DE Number: 7290380
zbMATH Keywords: prime ring
zbMATH Keywords: maximal right rings of quotients
zbMATH Keywords: extended Jacobson density theorem
zbMATH Keywords: generalized skew derivation
zbMATH Keywords: automorphism
MaRDI profile type: Publication
full work available at URL: s41980-019-00344-x
OpenAlex ID: W3004599150
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DOI: 10.1007/S41980-019-00344-X
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description: scientific article; zbMATH DE number 7290380