Results 141 to 150 of about 427 (170)

On “On derivations and commutativity of prime rings with involution”

Georgian Mathematical Journal, 2020
Abstract In this note, we indicate some errors in [S. Ali, N. A. Dar and M. Asci, On derivations and commutativity of prime rings with involution, Georgian Math. J. 23 2016, 1, 9–14] and present the correct versions of the erroneous results.
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Ideals and Higher Derivations in Commutative Rings

Canadian Journal of Mathematics, 1972
In this paper, we wish to generalize the following lemma first proven by O. Zariski [5, Lemma 4]. Let O be a complete local ring containing the rational numbers and let m denote the maximal ideal of O. Assume there exists a derivation δ of O such that δ(x) is a unit in O for some x in m.
Brown, William C., Kuan, Wei-Eihn
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\(\Phi\)-derivations and commutativity of rings and algebras

2022
Summary: The main purpose of this paper is to investigate the effect of \(\Phi \)-derivatives on the commutativity of rings and algebras. Let \(\mathfrak{R}\) be a 2-torsion free prime ring, \(d: \mathfrak{R} \rightarrow \mathfrak{R}\) be a \(\Phi\)-derivation such that \(\Phi\) is an epimorphism and \(d\Phi = \Phi d = d\). If \([\Phi (a), \Phi (x)]d(y)
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On derivations and commutativity in semiprime rings

Communications in Algebra, 1995
Let R be a ring, Z its center, U a nonzero left ideal, and D:R → R a derivation. We show that if R is semiprime with suitably-restricted additive torsion, then R must contain nonzero central ideals if one of the following holds: (i) [x, [x, D(x)]] ∊ Z for all x ∊ U; (ii) for a fixed positive integer n, [xn, D(x)] ∊ Z for all x ...
Qing Deng, Howard E. Bell
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On commutativity of rings with generalized derivations

Mathematical Journal of Okayama University, 2002
An additive map \(F\) from a ring \(R\) into itself is said to be a generalized derivation if there is a derivation \(d\) of \(R\) such that \(F(xy)=F(x)y+xd(y)\) for all \(x,y\in R\). The author extends some results that are known for derivations on prime rings to generalized derivations. The main results treat the conditions: (i) \([F(x),x]=0\), (ii)
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On derivations and commutativity in prime rings

Acta Mathematica Hungarica, 1995
Let \(R\) be a prime ring, \(U\) be a right ideal of \(R\), and \(d\) be a nonzero derivation of \(R\). It is shown that each of the following three conditions (i) \([d(x),d(y)] = d([y,x])\) for all \(x,y\in R\), (ii) \([d(x),d(y)] = d([x,y])\) for all \(x,y\in R\), (iii) \(\text{char\,}R\neq 2\) and \(d([x,y]) = 0\) for all \(x,y\in R\), implies that ...
Bell, H. E., Daif, M. N.
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On the annihilator of commutators with derivation in prime rings

Rendiconti del Circolo Matematico di Palermo, 2000
The main result of the paper is that for a prime ring \(R\) with \(\text{char\,}R\neq 2\), nonzero derivation \(D\) of \(R\), and noncentral Lie ideal \(L\) of \(R\), the left annihilator of \(\{[d(u),u]\mid u\in L\}\) is zero. The proof of the first part of the crucial Lemma 3 is incomplete, and incorrect as given.
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On commutativity of a factor ring R/P with derivations

ANNALI DELL'UNIVERSITA' DI FERRARA, 2023
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Derivations in commutators with power central values in rings

Publicationes Mathematicae Debrecen, 2010
Let \(R\) be a prime ring of characteristic not 2, let \(I\) be a nonzero ideal of \(R\), and let \(d\) be a nonzero derivation of \(R\). Suppose there exist positive integers \(n\) and \(k\) such that \([d(x^k),x^k]^n\) is central for every \(x\in I\). Then \(R\) satisfies the standard polynomial identity of degree \(4\).
Du, Yi-qiu, Wang, Yu
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On the higher derivations of commutative rings

Mathematical Journal of Okayama University, 1987
Let k be a commutative ring, A a commutative k-algebra, \(\{D_ r| r=0,1,2,...\}\) a higher derivation of A/k into A in the sense of Hasse and F. K. Schmidt (H-S sequence). The authors prove the following theorem, which is closely related to a conjecture of Nakai on high order derivations: If k is a field of characteristic zero then there is a unique ...
Abu-Saymeh, Sadi, Ikeda, Masatoshi
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