Affiliation:
1. College of Sciences, Northeast Electric Power University , Jilin 132012 , China
2. School of Mathematics and Statistics, Linyi University , Linyi 276005 , China
Abstract
Abstract
Let
R
R
be a unital associative ring. Our motivation is to prove that left derivations in column finite matrix rings over
R
R
are equal to zero and demonstrate that a left derivation
d
:
T
→
T
d:{\mathcal{T}}\to {\mathcal{T}}
in the infinite upper triangular matrix ring
T
{\mathcal{T}}
is determined by left derivations
d
j
{d}_{j}
in
R
(
j
=
1
,
2
,
…
)
R\left(j=1,2,\ldots )
satisfying
d
(
(
a
i
j
)
)
=
(
b
i
j
)
d\left(\left({a}_{ij}))=\left({b}_{ij})
for any
(
a
i
j
)
∈
T
\left({a}_{ij})\in {\mathcal{T}}
, where
b
i
j
=
d
j
(
a
11
)
,
i
=
1
,
0
,
i
≠
1
.
{b}_{ij}=\left\{\begin{array}{ll}{d}_{j}\left({a}_{11}),& i=1,\\ 0,& i\ne 1.\end{array}\right.
The similar results about Jordan left derivations are also obtained when
R
R
is 2-torsion free.