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Ω-Gorenstein Modules over Formal Triangular Matrix Rings
Let A and B be rings and U a ( B , A )-bimodule. Under some conditions, Ω -Gorenstein module over the formal triangular matrix ring T = A 0 U B is explicitly described, where Ω is a class of left T -modules. As an application, it is shown that if B U has finite projective dimension and U A has fini...
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| Published in: | Bulletin of the Malaysian Mathematical Sciences Society 2021-11, Vol.44 (6), p.4357-4366 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
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| container_end_page | 4366 |
| container_issue | 6 |
| container_start_page | 4357 |
| container_title | Bulletin of the Malaysian Mathematical Sciences Society |
| container_volume | 44 |
| creator | Wu, Dejun Yi, Chengang |
| description | Let
A
and
B
be rings and
U
a (
B
,
A
)-bimodule. Under some conditions,
Ω
-Gorenstein module over the formal triangular matrix ring
T
=
A
0
U
B
is explicitly described, where
Ω
is a class of left
T
-modules. As an application, it is shown that if
B
U
has finite projective dimension and
U
A
has finite flat dimension, then
M
=
M
1
M
2
φ
M
is a Gorenstein projective left
T
-module if and only if
M
1
is a Gorenstein projective left
A
-module,
Coker
(
φ
M
)
is a Gorenstein projective left
B
-module and
φ
M
:
U
⊗
A
M
1
→
M
2
is a monomorphism. This statement covers an earlier result of Enochs, Cortés-Izurdiaga and Torrecillas in this direction. |
| doi_str_mv | 10.1007/s40840-021-01169-w |
| format | article |
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A
and
B
be rings and
U
a (
B
,
A
)-bimodule. Under some conditions,
Ω
-Gorenstein module over the formal triangular matrix ring
T
=
A
0
U
B
is explicitly described, where
Ω
is a class of left
T
-modules. As an application, it is shown that if
B
U
has finite projective dimension and
U
A
has finite flat dimension, then
M
=
M
1
M
2
φ
M
is a Gorenstein projective left
T
-module if and only if
M
1
is a Gorenstein projective left
A
-module,
Coker
(
φ
M
)
is a Gorenstein projective left
B
-module and
φ
M
:
U
⊗
A
M
1
→
M
2
is a monomorphism. This statement covers an earlier result of Enochs, Cortés-Izurdiaga and Torrecillas in this direction.</description><identifier>ISSN: 0126-6705</identifier><identifier>EISSN: 2180-4206</identifier><identifier>DOI: 10.1007/s40840-021-01169-w</identifier><language>eng</language><publisher>Singapore: Springer Singapore</publisher><subject>Applications of Mathematics ; Mathematics ; Mathematics and Statistics ; Modules ; Rings (mathematics)</subject><ispartof>Bulletin of the Malaysian Mathematical Sciences Society, 2021-11, Vol.44 (6), p.4357-4366</ispartof><rights>Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2021</rights><rights>Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2021.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c270t-645c1972663ab645432b8d248c87edfa6a5f25a9ee9e5d7c8ab471e113de47023</cites><orcidid>0000-0001-6487-7416</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,27922,27923</link.rule.ids></links><search><creatorcontrib>Wu, Dejun</creatorcontrib><creatorcontrib>Yi, Chengang</creatorcontrib><title>Ω-Gorenstein Modules over Formal Triangular Matrix Rings</title><title>Bulletin of the Malaysian Mathematical Sciences Society</title><addtitle>Bull. Malays. Math. Sci. Soc</addtitle><description>Let
A
and
B
be rings and
U
a (
B
,
A
)-bimodule. Under some conditions,
Ω
-Gorenstein module over the formal triangular matrix ring
T
=
A
0
U
B
is explicitly described, where
Ω
is a class of left
T
-modules. As an application, it is shown that if
B
U
has finite projective dimension and
U
A
has finite flat dimension, then
M
=
M
1
M
2
φ
M
is a Gorenstein projective left
T
-module if and only if
M
1
is a Gorenstein projective left
A
-module,
Coker
(
φ
M
)
is a Gorenstein projective left
B
-module and
φ
M
:
U
⊗
A
M
1
→
M
2
is a monomorphism. This statement covers an earlier result of Enochs, Cortés-Izurdiaga and Torrecillas in this direction.</description><subject>Applications of Mathematics</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Modules</subject><subject>Rings (mathematics)</subject><issn>0126-6705</issn><issn>2180-4206</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><recordid>eNp9kNFKwzAUhoMoOOZewKuC19GTNE3SSxluChuCzOuQtqejo2tm0jp9JF_GZzJawTvPzeHA9_8HPkIuGVwzAHUTBGgBFDijwJjM6fGETDjTQAUHeUomwLikUkF2TmYh7CBOJrnkbELyzw-6dB670GPTJWtXDS2GxL2iTxbO722bbHxju-3QWp-sbe-bt-Sp6bbhgpzVtg04-91T8ry428zv6epx-TC_XdGSK-ipFFnJcsWlTG0RD5HyQldc6FIrrGorbVbzzOaIOWaVKrUthGLIWFqhUMDTKbkaew_evQwYerNzg-_iS8MzlStQSstI8ZEqvQvBY20Ovtlb_24YmG9LZrRkoiXzY8kcYygdQyHC3Rb9X_U_qS_61GoN</recordid><startdate>20211101</startdate><enddate>20211101</enddate><creator>Wu, Dejun</creator><creator>Yi, Chengang</creator><general>Springer Singapore</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0001-6487-7416</orcidid></search><sort><creationdate>20211101</creationdate><title>Ω-Gorenstein Modules over Formal Triangular Matrix Rings</title><author>Wu, Dejun ; Yi, Chengang</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c270t-645c1972663ab645432b8d248c87edfa6a5f25a9ee9e5d7c8ab471e113de47023</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Applications of Mathematics</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Modules</topic><topic>Rings (mathematics)</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Wu, Dejun</creatorcontrib><creatorcontrib>Yi, Chengang</creatorcontrib><collection>CrossRef</collection><jtitle>Bulletin of the Malaysian Mathematical Sciences Society</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Wu, Dejun</au><au>Yi, Chengang</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Ω-Gorenstein Modules over Formal Triangular Matrix Rings</atitle><jtitle>Bulletin of the Malaysian Mathematical Sciences Society</jtitle><stitle>Bull. Malays. Math. Sci. Soc</stitle><date>2021-11-01</date><risdate>2021</risdate><volume>44</volume><issue>6</issue><spage>4357</spage><epage>4366</epage><pages>4357-4366</pages><issn>0126-6705</issn><eissn>2180-4206</eissn><abstract>Let
A
and
B
be rings and
U
a (
B
,
A
)-bimodule. Under some conditions,
Ω
-Gorenstein module over the formal triangular matrix ring
T
=
A
0
U
B
is explicitly described, where
Ω
is a class of left
T
-modules. As an application, it is shown that if
B
U
has finite projective dimension and
U
A
has finite flat dimension, then
M
=
M
1
M
2
φ
M
is a Gorenstein projective left
T
-module if and only if
M
1
is a Gorenstein projective left
A
-module,
Coker
(
φ
M
)
is a Gorenstein projective left
B
-module and
φ
M
:
U
⊗
A
M
1
→
M
2
is a monomorphism. This statement covers an earlier result of Enochs, Cortés-Izurdiaga and Torrecillas in this direction.</abstract><cop>Singapore</cop><pub>Springer Singapore</pub><doi>10.1007/s40840-021-01169-w</doi><tpages>10</tpages><orcidid>https://orcid.org/0000-0001-6487-7416</orcidid></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0126-6705 |
| ispartof | Bulletin of the Malaysian Mathematical Sciences Society, 2021-11, Vol.44 (6), p.4357-4366 |
| issn | 0126-6705 2180-4206 |
| language | eng |
| recordid | cdi_proquest_journals_2579707786 |
| source | Springer Nature |
| subjects | Applications of Mathematics Mathematics Mathematics and Statistics Modules Rings (mathematics) |
| title | Ω-Gorenstein Modules over Formal Triangular Matrix Rings |
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