Affiliation:
1. Department of Mathematics, College of Natural and Computational Sciences, Ambo University , Ambo , Ethiopia
2. Department of Basic Disciplines, Chuzhou Polytechnic College , Chuzhou , Anhui , China
Abstract
Abstract
Let
Δ
=
A
N
B
A
M
A
B
B
\Delta =\left(\begin{array}{cc}A& {}_{A}N_{B}\\ {}_{B}M_{A}& B\end{array}\right)
be a Morita ring, where
M
⊗
A
N
=
0
=
N
⊗
B
M
M{\otimes }_{A}N=0=N{\otimes }_{B}M
. Let
X
X
be left
A
A
-module and
Y
Y
be left
B
B
-module. We prove that
(
X
,
M
⊗
A
X
,
1
,
0
)
⊕
(
N
⊗
B
Y
,
Y
,
0
,
1
)
\left(X,M{\otimes }_{A}X,1,0)\oplus \left(N{\otimes }_{B}Y,Y,0,1)
is a silting module if and only if
X
X
is a silting
A
A
-module,
Y
Y
is a silting
B
B
-module,
M
⊗
A
X
M{\otimes }_{A}X
is generated by
Y
Y
, and
N
⊗
B
Y
N{\otimes }_{B}Y
is generated by
X
X
. As a consequence, we obtain that if
M
A
{M}_{A}
and
N
B
{N}_{B}
are flat, then
(
X
,
M
⊗
A
X
,
1
,
0
)
⊕
(
N
⊗
B
Y
,
Y
,
0
,
1
)
\left(X,M{\otimes }_{A}X,1,0)\oplus \left(N{\otimes }_{B}Y,Y,0,1)
is a tilting
Δ
\Delta
-module if and only if
X
X
is a tilting
A
A
-module,
Y
Y
is a tilting
B
B
-module,
M
⊗
A
X
M{\otimes }_{A}X
is generated by
Y
Y
, and
N
⊗
B
Y
N{\otimes }_{B}Y
is generated by
X
X
.
Reference20 articles.
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4. I. Assem and N. Marmaridis, Tilting modules over split-by-nilpotent extensions, Comm. Algebra 26 (1998), no. 5, 1547–1555.
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