Let R be a noncommutative prime ring with extended centroid C and maximal right ring of quotients Q. Let I be a nonzero ideal of R, and let g, h be two generalized derivations of R. Suppose that
for ...all
where
are fixed positive integers. Then, one of the following conditions holds:
there exist
such that
and
for all
m = s, n = t and there exist
and
such that g(x) = xa and
for all
C is a finite field,
the
matrix ring over C for some integer
and there exist
such that g(x) = xa and h(x) = ux for all
Let R be a prime algebra over a commutative ring K, let I be a nonzero ideal of R and let
be a multilinear polynomial over K in k non-commuting indeterminates which is not central-valued on R. ...Suppose that
and d is a derivation of R such that
for all
, where
, s, t, m are fixed positive integers and
is the centre of R. It is shown that a = 0 or d = 0 except when
, the
matrix ring over a field F. This result gives a natural generalization of several well-known theorems in the literature. Moreover, with this we give an affirmative answer to the open conjecture recently raised by Huang in Derivations with annihilator conditions on Lie ideals in prime rings. J Algebra Appl. 2020;19:2050025.
In this article, the two-parameter quantum Heisenberg enveloping algebra, which serves as a model for certain quantum generalized Heisenberg algebras, has been studied at roots of unity. In this ...context, the quantum Heisenberg enveloping algebra becomes a polynomial identity algebra, and the dimension of simple modules is bounded by its PI degree. The PI degree, center, and complete classification of simple modules up to isomorphism are explicitly presented. We work over a field of arbitrary characteristic, although our results concerning the representations require that it is algebraically closed.
Let $\mathscr{R}$ be a prime ring of Char$(\mathscr{R}) \neq 2$ and $m\neq 1$
be a positive integer. If $S$ is a nonzero skew derivation with an associated
automorphism $\mathscr{T}$ of $\mathscr{R}$ ...such that $(S(a, b), a,
b)^{m} = S(a, b), a, b$ for all $a, b \in \mathscr{R}$, then
$\mathscr{R}$ is commutative.
For an arbitrary q≥2, we find an upper bound for the dimension of a subalgebra of the full matrix algebra Mn(K) over an arbitrary field K satisfying the identityx1,y1,z1⋅x2,y2,z2⋅⋯⋅xq,yq,zq=0, and we ...show that this upper bound is sharp by presenting an example in block triangular form of a subalgebra of Mn(K) with dimension equal to the obtained upper bound. We apply this result to Lie solvable algebras of index 2, i.e., algebras satisfying the identity x1,y1,x2,y2=0. To be precise, for n≤4, we find the sharp upper bound for the dimension of a Lie solvable subalgebra of Mn(K) of index 2, and for n>4, we obtain the relatively tight (at least for small values of n>4) interval2+⌊3n28⌋,2+⌊5n212⌋ for the maximum dimension of a Lie solvable subalgebra of Mn(K) of index 2, the exact value of which is not known.
Let m, n be two nonzero fixed positive integers, R a prime ring with the right Martindale quotient ring Q, G and H two generalized skew derivations of R with the same associated automorphism α of R. ...If
G
(
x
m
+
n
)
=
H
(
x
m
)
x
n
is an identity for R, then either R is commutative or there exists
q
∈
Q
such that
G
(
x
)
=
H
(
x
)
=
q
x
, for any
x
∈
R
.
Weak right η -centralizers of prime rings Siddeeque, Mohammad Aslam; Khan, Nazim; Abdullah, Ali Ahmed
Journal of Taibah University for Science,
12/2024, Letnik:
18, Številka:
1
Journal Article
Recenzirano
Odprti dostop
This work encompasses the study of certain weak maps termed as weak right η-centralizers defined on Formula: see text a noncommutative prime ring to Formula: see text. The first section provides an ...overview of the motivation behind our work. In the second section, our main focus is to establish the characterization of weak right η-centralizers while addressing the theory of functional identities, particularly in the challenging case of low dimensions. Furthermore, in the third section, we derive the characterization of weak Jordan σ-derivations based on the aforementioned characterization.
