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PureMathematics
n
Ø
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,2022,12(7),1160-1168
PublishedOnlineJuly2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.127127
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FP-
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r
GorensteinFP-
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StronglyGorensteinFP-InjectiveModules
overFormalTriangularMatrixRings
JinTan
CollegeofMathematicsandStatistics,NorthwestNormalUniversity, LanzhouGansu
Received:Jun.5
th
,2022;accepted:Jul.7
th
,2022;published:Jul.14
th
,2022
©
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n
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,2022,12(7):1160-1168.
DOI:10.12677/pm.2022.127127
?
Abstract
This paper considersstronglyGorensteinFP-injective modules over formal triangular
matrixrings.Let
T
=
A
0
UB
!
beformaltriangularmatrixring,where
A
and
B
are
tworingsand
U
isa
(
B,A
)
-bimodule.Itisprovedthatif
T
isaleftcoherentring,
B
U
is finitely presented and
pd
(
B
U
)
<
∞
,
M
=
M
1
M
2
!
ϕ
M
is strongly Gorenstein FP-injective
left
T
-modules,then
ker
g
ϕ
M
isstronglyGorensteinFP-injectiveleft
A
-modules,
M
2
is
stronglyGorensteinFP-injectiveleft
B
-modules,and
g
ϕ
M
isanepimorphism.
Keywords
FormalTriangularMatrixRing, FP-InjectiveModule,StronglyGorensteinFP-Injective
Module
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/pm.2022.1271271162
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DOI:10.12677/pm.2022.1271271163
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1
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(
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d
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/
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f
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/
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E
f
0
/
/
···
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1
0
!
∼
=
ker
f
0
!
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P
1
P
2
!
ϕ
P
´
?
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•
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,
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(
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2.5
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Q
g
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Q
g
/
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Q
g
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2
∼
=
ker(
g
),
¿
…
é
?
¿
k
•
L
«
†
B
-
P
…
pd
(
P
)
<
∞
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k
Hom
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(
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Φ)
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•
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U
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k
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,
…
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(
B
U
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¤
±
Hom
B
(
U,
Φ)
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,
=
k
†
A
Ü
S
Hom
B
(
U,
Φ) :
···
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∗
/
/
Hom
B
(
U,Q
)
g
∗
/
/
Hom
B
(
U,Q
)
g
∗
/
/
Hom
B
(
U,Q
)
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/
···
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Ï
d
†
T
-
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S
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···
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/
Hom
B
(
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Q
!
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∗
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Hom
B
(
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Q
!
g
∗
g
/
/
Hom
B
(
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)
Q
!
/
/
···
¦
Hom
B
(
U,M
2
)
M
2
!
∼
=
ker
g
∗
g
!
.
?
k
•
L
«
†
T
-
P
1
P
2
!
ϕ
P
,
…
pd
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P
1
P
2
!
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<
∞
.
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•
T
´
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¤
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d
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z
[9]
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•
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´
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v
à
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.
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n
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Ú
Ú
n
2.5
Œ
•
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2
´
k
•
L
«
,
…
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(
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2
)
<
∞
,
¤
±
Hom
B
(
P
2
,
Φ)
´
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.
d
q
,
h
¼
f
Š
‘
5
Œ
Hom
T
(
P
1
P
2
!
,
Hom
B
(
U,Q
)
Q
!
)
∼
=
Hom
B
(
P
2
,Q
),
¤
±
k
E
/
Ó
Hom
T
(
P
1
P
2
!
,
∆)
∼
=
Hom
B
(
P
2
,
Φ)
Ü
,
Ï
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Hom
B
(
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2
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M
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!
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GorensteinFP-
S
†
T
-
.
DOI:10.12677/pm.2022.1271271164
n
Ø
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n
3.2
M
=
M
1
M
2
!
ϕ
M
´
†
T
-
.
T
´
†
v
à
‚
,
B
U
´
k
•
L
«
,
…
pd
(
B
U
)
<
∞
.
X
J
M
=
M
1
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DOI:10.12677/pm.2022.1271271167
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,
Ü
}
±
.
/
ª
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Ý
‚
þ
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S
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I
|
,
•
B
.
/
ª
n
Ý
‚
þ
PC-
S
[J].
o
A
“
‰
Œ
ÆÆ
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,
‰
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DOI:10.12677/pm.2022.1271271168
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