Affiliation:
1. Department of Mathematics, Harbin University of Science and Technology , Harbin 150080 , China
Abstract
Abstract
The aim of this paper is to study the structure of arbitrary split twisted inner derivation triple systems. We obtain a sufficient condition for the decomposition of arbitrary twisted inner derivation triple system
T
{\mathscr{T}}
which is of the form
T
=
U
+
∑
[
θ
]
∈
Λ
T
/
∼
I
[
θ
]
{\mathscr{T}}=U+{\sum }_{\left[\theta ]\in {\Lambda }^{{\mathscr{T}}}\text{/} \sim }{I}_{\left[\theta ]}
with
U
U
a subspace of
T
0
{{\mathscr{T}}}_{0}
and any
I
[
θ
]
{I}_{\left[\theta ]}
a well-described ideal of
T
{\mathscr{T}}
, satisfying
{
I
[
θ
]
,
T
,
I
[
η
]
}
\left\{{I}_{\left[\theta ]},{\mathscr{T}},{I}_{\left[\eta ]}\right\}
=
{
I
[
θ
]
,
I
[
η
]
,
T
}
\left\{{I}_{\left[\theta ]},{I}_{\left[\eta ]},{\mathscr{T}}\right\}
=
{
T
,
I
[
θ
]
,
I
[
η
]
}
\left\{{\mathscr{T}},{I}_{\left[\theta ]},{I}_{\left[\eta ]}\right\}
=
{
I
[
θ
]
,
T
,
I
[
η
]
}
′
\left\{{I}_{\left[\theta ]},{\mathscr{T}},{I}_{\left[\eta ]}\right\}^{\prime}
=
{
I
[
θ
]
,
I
[
η
]
,
T
}
′
\left\{{I}_{\left[\theta ]},{I}_{\left[\eta ]},{\mathscr{T}}\right\}^{\prime}
=
{
T
,
I
[
θ
]
,
I
[
η
]
}
′
=
0
\left\{{\mathscr{T}},{I}_{\left[\theta ]},{I}_{\left[\eta ]}\right\}^{\prime} =0
if
[
θ
]
≠
[
η
]
\left[\theta ]\ne \left[\eta ]
. In particular, a necessary and sufficient condition for the simplicity of the triple system is given.