Let A be a noncommutative unital prime algebra and let S be a commutative unital algebra over a field F. We describe the form of biderivations of the algebra A⊗S. As an application, we determine the ...form of commuting linear maps of A⊗S.
Let R be a triangular ring. The problem of describing the form of additive maps F1,F2,G1,G2:R→R satisfying functional identity F1(x)y+F2(y)x+xG2(y)+yG1(x)=0 for all x,y∈R is considered. As an ...application generalized inner biderivations and commuting additive maps of certain triangular rings are determined.
Let R be a subring of a ring Q, both having the same unity. We prove that if R is a d-free subset of Q, then the upper triangular matrix ring Tn(R) is a d-free subset of Tn(Q) for any n∈N.
Using the notion of the maximal left ring of quotients, our recent result on the solutions of functional identity
in triangular rings is generalized. Consequently, generalizations of known results on ...commuting additive maps and generalized inner biderivations of triangular rings are obtained.
Let
T
n
(
R
)
be the upper triangular matrix ring over a unital ring
R
. Suppose that
B
:
T
n
(
R
)
×
T
n
(
R
)
→
T
n
(
R
)
is a biadditive map such that
B
(
X
,
X
)
X
=
X
B
(
X
,
X
)
for all
X
∈
T
n
...(
R
)
. We consider the problem of describing the form of the map
X
↦
B
(
X
,
X
)
.
We consider the problem of describing the form of biderivations of a triangular ring. Our approach is based on the notion of the maximal left ring of quotients, which enables us to generalize ...Benkovič’s result on biderivations (Benkovič in Linear Algebra Appl 431:1587–1602,
2009
). Our result is applied to block upper triangular matrix rings and nest algebras.
We introduce the notion of a multiplicative Lie n-derivation of a ring, generalizing the notion of a Lie (triple) derivation. The main goal of the paper is to consider the question of when do all ...multiplicative Lie n-derivations of a triangular ring T have the so-called standard form. The main result is applied to the classical examples of triangular rings: nest algebras and (block) upper triangular matrix rings.
Let
A
be a triangular algebra. The problem of describing the form of a bilinear map
B
:
A
×
A
→
A
satisfying
B
(
x
,
x
)
x
=
x
B
(
x
,
x
)
for all
x
∈
A
is considered. As an application, ...commutativity preserving maps and Lie isomorphisms of certain triangular algebras (e.g., upper triangular matrix algebras and nest algebras) are determined.
Let
R
be a semiprime ring with the maximal right ring of quotients
Q
mr
. An additive map
d
:
R
→
Q
mr
is called a generalized skew derivation if there exists a ring endomorphism
σ
:
R
→
R
and a map
...such that
for all
x
,
y
∈
R
. If
σ
is surjective, we determine the structure of generalized skew derivations for which there exists a finite number of elements
a
i
,
b
i
∈
Q
mr
such that
d
(
x
) =
a
1
xb
1
+ ⋯ +
a
n
xb
n
for all
x
∈
R
.
PDF
