Let
be a ∗−prime ring with characteristic not 2,
a nonzero ∗− (
)−Lie ideal of
,
a nonzero derivation of
. Suppose
,
be two automorphisms of
such that
=
,
=
and ∗ commutes with
,
,
. In the present ...paper it is shown that if
) ⊆
or
) ⊆
, then
⊆
Let
R
be a prime ring of characteristic different from 2 with center
Z
and extended centroid
C
, and let
L
be a Lie ideal of
R
. Consider two nontrivial automorphisms
α
and
β
of
R
for which there ...exist integers
m,n
≥ 1 such that
α
(
u
)
n
+
β
(
u
)
m
= 0 for all
u
∈
L
. It is shown that, under these assumptions, either
L
is central or
R
⊆
M
2
(
C
) (where
M
2
(
C
) is the ring of 2 × 2 matrices over
C
),
L
is commutative, and
u
2
∈
Z
for all
u
∈
L
. In particular, if
L
=
R, R
, then
R
is commutative.
Let n≥1 be a fixed integer, R a prime ring with its right Martindale quotient ring Q, C the extended centroid, and L a non-central Lie ideal of R. If F is a generalized skew derivation of R such that ...(F(x)F(y)−yx)
n
= 0 for all x,y∈L, then char(R) = 2 and R⊆M
2
(C), the ring of 2×2 matrices over C.
Let $R$ be a prime ring, $U$ the Utumi quotient ring of $R,$ $C$ the extended centroid of $R$ and $L$ a noncentral Lie ideal of $R.$ If $R$admits a generalized derivation $F$ associated with a ...derivation $\delta$ of $R$ such that for some fixed integers $m,n\geq 1,$ $F(u,v)^{m}=u,v_{n}$ for all $u,v\in L,$ then one of the following holds true:
(i) $R$ satisfies $s_{4},$ the standard identity in four variables.
(ii) there exists $\lambda\in C$ such that $F(x)=\lambda x$ for all $x\in R.$ Moreover, if $n=1,$ then $\lambda^{m}=1$ and if $n>1,$ then $F=0.$
Let
R
be a non-commutative prime ring,
Z
(
R
) its center,
Q
its right Martindale quotient ring,
C
its extended centroid,
F
≠
0
an
b
-generalized skew derivation of
R
,
L
a non-central Lie ideal of
R
...,
0
≠
a
∈
R
and
n
≥
1
a fixed integer. In this paper, we prove the following two results:
If
R
has characteristic different from 2 and 3 and
a
F
(
x
)
,
x
n
=
0
, for all
x
∈
L
, then either there exists an element
λ
∈
C
, such that
F
(
x
)
=
λ
x
, for all
x
∈
R
or
R
satisfies
s
4
(
x
1
,
…
,
x
4
)
, the standard identity of degree 4, and there exist
λ
∈
C
and
b
∈
Q
, such that
F
(
x
)
=
b
x
+
x
b
+
λ
x
, for all
x
∈
R
.
If
char
(
R
)
=
0
or
char
(
R
)
>
n
and
a
F
(
x
)
,
x
n
∈
Z
(
R
)
, for all
x
∈
R
, then either there exists an element
λ
∈
C
, such that
F
(
x
)
=
λ
x
, for all
x
∈
R
or
R
satisfies
s
4
(
x
1
,
…
,
x
4
)
.
In the given study, we intended to gain familiarity with the idea of fuzzy Hom–Lie subalgebras (ideals) of Hom–Lie algebras. It primarily seeks to study a few of their properties. This research ...investigates the relationship between fuzzy Hom–Lie subalgebras (ideals) and Hom–Lie subalgebras (ideals). Additionally, this study constructs new fuzzy Hom–Lie subalgebras based on the direct sum of a finite number of existing ones. Finally, the properties of fuzzy Hom–Lie subalgebras and fuzzy Hom–Lie ideals are examined in the context of the morphisms of Hom–Lie algebras.
Let R be a prime ring with characteristic not 2 and ?,?,?,?,?,?,?
automorphisms of R. Let h : R ? R be a nonzero
left(resp.right)-generalized (?,?)-derivation, b ? R and V ? 0 a left (?,?)-Lie ideal ...of R. The main object in this article is to study the
situations. (1) h(I) ? C?,?(V),(2) bh(I) ? C?,?(V) or h(I)b ? C?,?0V), (3)
h?(V)=0,(4) h?(V)b=0 or bh?(V)=0.
nema
Let H be an in_nite dimensional separable Hilbert space with a fixed orthonormal base {e 1 , e 2 , …}. Let L be the subspace lattice generated by the subspaces {e 1 , e 1 , e 2 , e 1 , e 2 , e 3 , …} ...and let AlgL be the algebra of bounded operators which leave invariant all projections in L. Let p and q be natural numbers(p ≤ q). Let B p,q = { T ∈ AlgL | T (p,q) = 0 }. Let A be a linear manifold in AlgL such that {0} □ A ⊂ B p,q . If A is an ideal in AlgL, then T (I,j) = 0; p ≤ i 6 q and i ≤ j ≤ q for all T in A.
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