Derivations in Prime Rings

1983 ◽  
Vol 26 (3) ◽  
pp. 267-270 ◽  
Author(s):  
Jeffrey Bergen
Keyword(s):  
Commutative Ring ◽  
Prime Ring ◽  
Simple Algebra ◽  
Finitely Generated ◽  
Prime Rings ◽  
Finite Dimensional ◽  
The Relationship

AbstractLet R be a prime ring and d≠0 a derivation of R. We examine the relationship between the structure of R and that of d(R). We prove that if R is an algebra over a commutative ring A such that d(R) is a finitely generated submodule then R is an order in a simple algebra finite dimensional over its center.

m-potent commutators involving skew derivations and multilinear polynomials

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohammad Ashraf ◽  
Sajad Ahmad Pary ◽  
Mohd Arif Raza
Keyword(s):  
Prime Ring ◽  
Quotient Ring ◽  
Multilinear Polynomial ◽  
Extended Centroid ◽  
Prime Rings ◽  
Skew Derivation ◽  
Martindale Quotient Ring ◽  
Multilinear Polynomials ◽  
The Right ◽  
The Relationship

AbstractLet {\mathscr{R}} be a prime ring, {\mathscr{Q}_{r}} the right Martindale quotient ring of {\mathscr{R}} and {\mathscr{C}} the extended centroid of {\mathscr{R}}. In this paper, we discuss the relationship between the structure of prime rings and the behavior of skew derivations on multilinear polynomials. More precisely, we investigate the m-potent commutators of skew derivations involving multilinear polynomials, i.e.,\big{(}[\delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})]\big{)}^{m}=[% \delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})],where {1<m\in\mathbb{Z}^{+}}, {f(x_{1},x_{2},\ldots,x_{n})} is a non-central multilinear polynomial over {\mathscr{C}} and δ is a skew derivation of {\mathscr{R}}.

Nilpotent Inner Derivations of the Skew Elements of Prime Rings With Involution

1991 ◽  
Vol 43 (5) ◽  
pp. 1045-1054 ◽  
Author(s):  
W. S. Martindale ◽  
C. Robert Miers
Keyword(s):  
Prime Ring ◽  
Central Simple Algebra ◽  
Simple Algebra ◽  
Extended Centroid ◽  
Prime Rings ◽  
Rings With Involution ◽  
Central Simple

AbstractLet R be a prime ring with invoution *, of characteristic 0, with skew elements K and extended centroid C. Let a ∈ K be such that (ad a)n =0 on K. It is shown that one of the following possibilities holds: (a) R is an order in a 4-dimensional central simple algebra, (b) there is a skew element λ in C such that , (c) * is of the first kind, n ≡ 0 or n ≡ 3 (mod 4), and . Examples are given illustrating (c).

Derivations in differentially prime rings

2018 ◽  
Vol 17 (07) ◽  
pp. 1850129
Author(s):  
Ahmad Al Khalaf ◽  
Orest D. Artemovych ◽  
Iman Taha
Keyword(s):  
Commutative Ring ◽  
Prime Ring ◽  
Prime Rings ◽  
Commutative Case ◽  
Lie Ring ◽  
Lie Rings

Earlier properties of Lie rings [Formula: see text] of derivations in commutative differentially prime rings [Formula: see text] was investigated by many authors. We study Lie rings [Formula: see text] in the non-commutative case and shown that if [Formula: see text] is a [Formula: see text]-prime ring of characteristic [Formula: see text], then [Formula: see text] is a prime Lie ring or [Formula: see text] is a commutative ring.

A Theorem on Derivations of Prime Rings with Involution

1981 ◽  
Vol 34 (2) ◽  
pp. 356-369 ◽  
Author(s):  
I. N. Herstein
Keyword(s):  
Prime Ring ◽  
Simple Algebra ◽  
Prime Rings ◽  
Rings With Involution ◽  
Characteristic 2

In a recent note [2] we showed that if R is a prime ring and d ≠ 0 a derivation of R such that d(x)d(y) = d(y)d(x) for all x, y ∈ R then, if R is not a characteristic 2, R must be commutative. (If char R = 2 we showed that R must be an order in a 4-dimensional simple algebra.)In this paper we shall consider a similar problem, namely, that of a prime ring R with involution * where d(x)d(y) = d(y)d(x) not for all x, y ∈ R but merely for symmetric elements x* = x and y* = y. Although it is clear that some results can be obtained if R is of characteristic 2, we shall only be concerned with the case char R ≠ 2. Even in this case one cannot hope to extend the result cited in the first paragraph, that is, to show that R is commutative.

On Derivations in Prime Rings and a Question of Herstein

1979 ◽  
Vol 22 (3) ◽  
pp. 339-344 ◽  
Author(s):  
Amos Kovacs
Keyword(s):  
Prime Ring ◽  
Simple Algebra ◽  
Prime Rings

In [2], Herstein proves the following result:Theorem. Let R be a prime ring, d≠0 a derivation of R such that d(x) d(y) = d(y) d(x) for all x, y ∈ R. Then, if char r≠2, R is commutative, and if char R = 2, R is commutative or an order in a simple algebra which is 4-dimensional over its center.

Nilpotent-by-abelian Lie algebras of type FPm

2010 ◽  
Vol 148 (3) ◽  
pp. 429-437
Author(s):  
J. R. J. GROVES ◽  
DESSISLAVA H. KOCHLOUKOVA
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Finitely Generated ◽  
Nilpotent Lie Algebra ◽  
Tensor Power ◽  
Diagonal Action ◽  
Finite Dimensional ◽  
Commutator Subalgebra ◽  
Finiteness Properties ◽  
The Relationship

AbstractLet L be a finitely generated Lie algebra which is a split extension of a free nilpotent Lie algebra N by a finite dimensional abelian Lie algebra. Let V denote the quotient of N by its commutator subalgebra; we can regard V as a module for L/N. We discuss the relationship between the homological finiteness properties of V and those of L. In particular, we show that if L has type FPm and N has class c then V is 1 + c(m − 1)-tame (equivalently, the (1 + c(m − 1))th tensor power of V is finitely generated under the diagonal action of L/N).

scholarly journals PRIME RINGS WITH FINITENESS PROPERTIES ON ONE-SIDED IDEALS

2002 ◽  
Vol 45 (2) ◽  
pp. 507-511 ◽  
Author(s):  
Tsiu-Kwen Lee
Keyword(s):  
Prime Ring ◽  
Constant Term ◽  
Extended Centroid ◽  
Additive Subgroup ◽  
Prime Rings ◽  
Finiteness Conditions ◽  
Finite Dimensional ◽  
Finite Set ◽  
Primary 16N60 ◽  
Finiteness Properties

AbstractLet R be a prime ring with extended centroid C, $\rho$ a non-zero right ideal of R and let $f(X_1,\dots,X_t)$ be a polynomial, having no constant term, over C. Suppose that $f(X_1,\dots,X_t)$ is not central-valued on RC. We denote by $f(\rho)$ the additive subgroup of RC> generated by all elements $f(x_1,\dots,x_t)$ for $x_i\in\rho$. The main goals of this note are to prove two results concerning the extension properties of finiteness conditions as follows.(I) If $f(\rho)$ spans a non-zero finite-dimensional $C$-subspace of $RC$, then $\dim_CRC$ is finite.(II) If $f(\rho)\ne0$ and is a finite set, then $R$ itself is a finite ring.AMS 2000 Mathematics subject classification: Primary 16N60; 16R50

A note on Lie ideals of prime rings

2019 ◽  
Vol 18 (03) ◽  
pp. 1950045
Author(s):  
Tsiu-Kwen Lee ◽  
Muzibur Rahman Mozumder
Keyword(s):  
Division Algebras ◽  
Finitely Generated ◽  
Prime Rings ◽  
Central Division ◽  
Arbitrary Characteristic ◽  
Finite Dimensional

In this paper, we characterize Lie ideals, which are either finitely generated [Formula: see text]-modules or maximal, in (centrally closed) prime rings. As consequences, we extend the results proved in [1] for finite dimensional central division algebras of characteristic not [Formula: see text] to simple rings of arbitrary characteristic.

Central *-Differential Identities in Prime Rings

1996 ◽  
Vol 39 (2) ◽  
pp. 211-215
Author(s):  
P. H. Lee ◽  
T. L. Wong
Keyword(s):  
Prime Ring ◽  
Integral Domain ◽  
Central Simple Algebra ◽  
Simple Algebra ◽  
Prime Rings ◽  
Differential Identities ◽  
Central Simple

AbstractLet R be a prime ring with involution and d, δ be derivations on R. Suppose that xd(x)—δ(x)x is central for all symmetric x or for all skew x. Then d = δ = 0 unless R is a commutative integral domain or an order of a 4-dimensional central simple algebra.

Derivations of differentially semiprime rings

2019 ◽  
Vol 12 (05) ◽  
pp. 1950079
Author(s):  
Ahmad Al Khalaf ◽  
Iman Taha ◽  
Orest D. Artemovych ◽  
Abdullah Aljouiiee
Keyword(s):  
Commutative Ring ◽  
Semiprime Ring ◽  
Prime Rings ◽  
Commutative Case ◽  
Lie Ring ◽  
Lie Rings ◽  
Semiprime Rings ◽  
Torsion Free

Earlier D. A. Jordan, C. R. Jordan and D. S. Passman have investigated the properties of Lie rings Der [Formula: see text] of derivations in a commutative differentially prime rings [Formula: see text]. We study Lie rings Der [Formula: see text] in the non-commutative case and prove that if [Formula: see text] is a [Formula: see text]-torsion-free [Formula: see text]-semiprime ring, then [Formula: see text] is a semiprime Lie ring or [Formula: see text] is a commutative ring.

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