The aim of this paper is to give an improvement of a result on functional identities in upper triangular matrix rings obtained by Eremita, which presents a short proof of Eremita's result.
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.
Let
be a subset of a unital ring
such that 0 ∈
. Let us fix an element
∈
. If
is a (
;
)-free subset of
, then
) is a (
′;
)-free subset of
), where
′ ∈
),
=
,
= 1, 2, …,
, for any
∈
Let
A
=
M
n
(
F) be the matrix algebra over a field
F with an involution ∗, where
n
⩾
20. Suppose that
θ
:
A
→
A is a bijective linear map such that
θ(
x)
θ(
y)
=
θ(
y)
θ(
x)* for all
x,
y
∈
A such ...that
xy
=
yx*. We show that
θ is of the form
θ(
x)
=
λuxu
−1 for
x
∈
A, where
λ is a nonzero symmetric scalar and
u is a normal matrix such that
uu* is a nonzero scalar.
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