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n -Ding modules over formal lower triangular matrix rings- Acta Scientiarum Naturalium Universitatis Sunyatseni Vol. 61, Issue 4, Pages: 151-159(2022)
- 作者机构：
西北师范大学数学与统计学院， 甘肃 兰州 730070
- 作者简介：
- 基金信息：
DOI：10.13471/j.cnki.acta.snus.2020A023
CLC： O153.3Published：25 July 2022，
Published Online：07 January 2022，
Received：22 May 2020，
Accepted：05 March 2021
扫 描 看 全 文
张文汇,刘婷.形式下三角矩阵环上的n-Ding模[J].中山大学学报(自然科学版),2022,61(04):151-159.
ZHANG Wenhui,LIU Ting.n-Ding modules over formal lower triangular matrix rings[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2022,61(04):151-159.
张文汇,刘婷.形式下三角矩阵环上的n-Ding模[J].中山大学学报(自然科学版),2022,61(04):151-159. DOI： 10.13471/j.cnki.acta.snus.2020A023.
ZHANG Wenhui,LIU Ting.n-Ding modules over formal lower triangular matrix rings[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2022,61(04):151-159. DOI： 10.13471/j.cnki.acta.snus.2020A023.
摘要
讨论形式下三角矩阵环
<math id="M1"><mi>T</mi><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mi>A</mi></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mi>U</mi></mtd><mtd><mi>B</mi></mtd></mtr></mtable></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49449710&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49449687&type=
15.49400043
7.11199999
上的
<math id="M2"><mi>n</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451351&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451335&type=
1.60866666
2.28600001
-Ding 模（其中
<math id="M3"><mi>A</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49449785&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49449769&type=
1.77800000
2.28600001
，
<math id="M4"><mi>B</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49449818&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49449792&type=
1.94733346
2.28600001
是环，
<math id="M5"><mi>U</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49449848&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49449833&type=
2.11666679
2.28600001
是
<math id="M6"><mo stretchy="false">(</mo><mi>B</mi><mo>
</mo><mi>A</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49449900&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49449861&type=
7.70466709
2.96333337
-双模）。证明了：（i）设
<math id="M7"><msub><mrow><mi>U</mi></mrow><mrow><mi>A</mi></mrow></msub></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450351&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450336&type=
3.38666677
3.21733332
有有限的平坦维数，
<math id="M8"><mmultiscripts><mrow/><mprescripts/><mrow><mi>B</mi></mrow><mrow/></mmultiscripts><mi>U</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450314&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450321&type=
3.47133350
3.21733332
是平坦模。若
<math id="M9"><mi>M</mi><mo>=</mo><msub><mrow><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><msup><mrow><mi>φ</mi></mrow><mrow><mi>M</mi></mrow></msup></mrow></msub></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49449993&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49449975&type=
15.23999977
7.87400007
是
<math id="M10"><mi>n</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451351&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451335&type=
1.60866666
2.28600001
-Ding 投射左
<math id="M11"><mi>T</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450398&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450389&type=
1.77800000
2.28600001
-模，则
<math id="M12"><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450071&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450062&type=
3.64066648
3.21733332
是
<math id="M13"><mo stretchy="false">(</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450455&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450433&type=
9.22866726
2.96333337
-Ding 投射左
<math id="M14"><mi>A</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450465&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450452&type=
1.77800000
2.28600001
-模，
<math id="M15"><msup><mrow><mi>φ</mi></mrow><mrow><mi>M</mi></mrow></msup></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450183&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450163&type=
3.47133350
3.04800010
是单同态，并且
<math id="M16"><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>/</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450213&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450177&type=
4.82600021
3.21733332
<math id="M17"><mi mathvariant="normal">I</mi><mi mathvariant="normal">m</mi><mtext> </mtext><msup><mrow><mi>φ</mi></mrow><mrow><mi>M</mi></mrow></msup></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450232&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450221&type=
7.45066643
3.13266683
是
<math id="M18"><mi>n</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451351&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451335&type=
1.60866666
2.28600001
-Ding 投射左
<math id="M19"><mi>B</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450576&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450565&type=
1.94733346
2.28600001
-模；（ii）设
<math id="M20"><mi>T</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450302&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450284&type=
1.77800000
2.28600001
是右凝聚环，
<math id="M21"><mmultiscripts><mrow/><mprescripts/><mrow><mi>B</mi></mrow><mrow/></mmultiscripts><mi>U</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450314&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450321&type=
3.47133350
3.21733332
是平坦模，
<math id="M22"><msub><mrow><mi>U</mi></mrow><mrow><mi>A</mi></mrow></msub></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450351&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450336&type=
3.38666677
3.21733332
是有限表示模且有有限的投射维数。若
<math id="M23"><mi>W</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>
</mo><msub><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mi>W</mi></mrow></msub></mrow></msub></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450361&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450356&type=
19.72733307
4.74133301
是
<math id="M24"><mi>n</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451351&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451335&type=
1.60866666
2.28600001
-Ding 内射右
<math id="M25"><mi>T</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450398&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450389&type=
1.77800000
2.28600001
-模，则
<math id="M26"><msub><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450428&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450405&type=
3.64066648
3.21733332
是
<math id="M27"><mo stretchy="false">(</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450455&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450433&type=
9.22866726
2.96333337
-Ding 内射右
<math id="M28"><mi>A</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450465&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450452&type=
1.77800000
2.28600001
-模，
<math id="M29"><mover accent="true"><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mi>W</mi></mrow></msub></mrow><mo>̃</mo></mover></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450495&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450480&type=
3.55599999
3.97933316
是满同态，并且
<math id="M30"><mi mathvariant="normal">K</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi><mover accent="true"><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mi>W</mi></mrow></msub></mrow><mo>̃</mo></mover></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450532&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450517&type=
8.80533314
3.97933316
是
<math id="M31"><mi>n</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451351&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451335&type=
1.60866666
2.28600001
-Ding 内射右
<math id="M32"><mi>B</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450576&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450565&type=
1.94733346
2.28600001
-模。
Abstract
<math id="M33"><mi>n</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451398&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451390&type=
1.86266661
2.62466669
-Ding modules are investigated under the formal lower triangular matrix ring
<math id="M34"><mi>T</mi><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mi>A</mi></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mi>U</mi></mtd><mtd><mi>B</mi></mtd></mtr></mtable></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450623&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450612&type=
18.28800011
8.21266651
， where
<math id="M35"><mi>A</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450648&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450628&type=
2.28600001
2.62466669
and
<math id="M36"><mi>B</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450662&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450657&type=
2.28600001
2.62466669
are rings，
<math id="M37"><mi>U</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450688&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450682&type=
2.70933342
2.62466669
is a
<math id="M38"><mo stretchy="false">(</mo><mi>B</mi><mo>
</mo><mi>A</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450702&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450710&type=
9.65200043
3.47133350
-bimodule. It is proved that （i） If
<math id="M39"><msub><mrow><mi>U</mi></mrow><mrow><mi>A</mi></mrow></msub></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451073&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451052&type=
4.06400013
3.72533321
has finite flat dimension，
<math id="M40"><mmultiscripts><mrow/><mprescripts/><mrow><mi>B</mi></mrow><mrow/></mmultiscripts><mi>U</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450731&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450743&type=
4.06400013
3.72533321
is flat and
<math id="M41"><mi>M</mi><mo>=</mo><msub><mrow><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><msup><mrow><mi>φ</mi></mrow><mrow><mi>M</mi></mrow></msup></mrow></msub></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450770&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450735&type=
17.78000069
9.22866726
is a
<math id="M42"><mi>n</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451398&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451390&type=
1.86266661
2.62466669
-Ding projective left
<math id="M43"><mi>T</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451134&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451132&type=
2.11666679
2.62466669
-module， then
<math id="M44"><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450828&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450819&type=
4.31799984
3.72533321
is a
<math id="M45"><mo stretchy="false">(</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450838&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450833&type=
10.32933331
3.38666677
-Ding projective left
<math id="M46"><mi>A</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451206&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451186&type=
2.28600001
2.62466669
-module，
<math id="M47"><msup><mrow><mi>φ</mi></mrow><mrow><mi>M</mi></mrow></msup></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450907&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450888&type=
3.72533321
3.72533321
is a monomorphism， and
<math id="M48"><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>/</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450933&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450913&type=
5.67266655
3.72533321
<math id="M49"><mi mathvariant="normal">I</mi><mi mathvariant="normal">m</mi><mtext> </mtext><msup><mrow><mi>φ</mi></mrow><mrow><mi>M</mi></mrow></msup></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450950&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450923&type=
8.63599968
3.72533321
is a
<math id="M50"><mi>n</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451398&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451390&type=
1.86266661
2.62466669
-Ding projective left
<math id="M51"><mi>B</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451307&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451309&type=
2.28600001
2.62466669
-module；（ii） If
<math id="M52"><mi>T</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451014&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49450990&type=
2.11666679
2.62466669
is a right coherent ring，
<math id="M53"><mmultiscripts><mrow/><mprescripts/><mrow><mi>B</mi></mrow><mrow/></mmultiscripts><mi>U</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451046&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451031&type=
4.06400013
3.72533321
is flat，
<math id="M54"><msub><mrow><mi>U</mi></mrow><mrow><mi>A</mi></mrow></msub></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451073&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451052&type=
4.06400013
3.72533321
is finitely presented and has finite projective dimension and
<math id="M55"><mi>W</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>
</mo><msub><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mi>W</mi></mrow></msub></mrow></msub></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451099&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451086&type=
23.45266724
5.16466665
is a
<math id="M56"><mi>n</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451398&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451390&type=
1.86266661
2.62466669
-Ding injective right
<math id="M57"><mi>T</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451134&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451132&type=
2.11666679
2.62466669
-module， then
<math id="M58"><msub><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451156&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451151&type=
4.31799984
3.72533321
is a
<math id="M59"><mo stretchy="false">(</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451182&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451175&type=
10.32933331
3.38666677
-Ding injective right
<math id="M60"><mi>A</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451206&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451186&type=
2.28600001
2.62466669
-module，
<math id="M61"><mover accent="true"><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mi>W</mi></mrow></msub></mrow><mo>̃</mo></mover></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451240&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451222&type=
3.89466691
4.48733330
is an epimorphism， and
<math id="M62"><mi mathvariant="normal">K</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi><mover accent="true"><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mi>W</mi></mrow></msub></mrow><mo>̃</mo></mover></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451270&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451253&type=
10.15999985
4.48733330
is a
<math id="M63"><mi>n</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451398&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451390&type=
1.86266661
2.62466669
-Ding injective right
<math id="M64"><mi>B</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451307&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451309&type=
2.28600001
2.62466669
-module.
关键词
形式下三角矩阵环<math id="M65"><mi>n</mi></math>https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451351&type=https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451335&type=1.608666662.28600001-Ding投射模<math id="M66"><mi>n</mi></math>https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451351&type=https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451335&type=1.608666662.28600001-Ding内射模
Keywords
formal lower triangular matrix ring<math id="M67"><mi>n</mi></math>https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451398&type=https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451390&type=1.862666612.62466669-Ding projective module<math id="M68"><mi>n</mi></math>https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451398&type=https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49451390&type=1.862666612.62466669-Ding injective module
references
ENOCHS E E，JENDA O M G. Gorenstein injective and projective modules ［J］. Mathematische Zeitschrift， 1995， 220（4）： 611-633.
ENOCHS E E，JENDA O M G. Relative homological algebra ［M］. Berlin： Walter de Gruyter， 2000.
MAO L X，DING N Q. Gorenstein FP-injective and Gorenstein flat modules ［J］. Journal of Algebra and Its Applications， 2008， 7（4）： 491-506.
DING N Q，LI Y L，MAO L X. Strongly Gorenstein flat modules ［J］. Journal of Australian Mathematical Society， 2009， 86（3）： 323-338.
GILLESPIE J. Model structures on modules over Ding-Chen rings ［J］. Homology， Homotopy and Applications， 2010， 12（1）： 61-73.
TANG X. Applications of n-Gorenstein projective and injective modules ［J］.Hacettepe Journal of Mathematics and Statistics， 2015， 44（6）： 1435-1443.
HAGHANY A，VARADARAJAN K. Study of formal triangular matrix rings ［J］. Journal of Communications in Algebra， 1999， 27（11）： 5507-5525.
HAGHANY A，VARADARAJAN K. Study of modules over formal triangular matrix rings ［J］. Journal of Pure and Applied Algebra， 2000， 147（1）： 41-58.
ZHANG P. Gorenstein-projective modules and symmetric recollements ［J］. Journal of Algebra， 2013， 388： 65-80.
ENOCHS E E，CORTÉS-IZURDIAGA M，TORRECILLAS B. Gorenstein conditions over triangular matrix rings ［J］. Journal of Pure and Applied Algebra， 2014， 218（8）： 1544-1554.
MAO L X. Ding modules and dimensions over formal triangular matrix rings ［OL］.https：//arxiv.org/abs/1912.06968https://arxiv.org/abs/1912.06968.
GREEN E L. On the representation theory of rings in matrix form ［J］. Pacific Journal of Mathemtics， 1982， 100（1）： 123-138.
张铭，张文汇. n-Ding 投射和n-Ding内射模 ［J］. 山东大学学报（理学版）， 2019， 54（4）： 60-66.
FOSSUM R M，GRIFFITH P，REITEN I. Trivial extensions of Abelian categories ［M］. Berlin： Springer-Verlag， 1975.
ASADOLLAHI J，SALARIAN S. On the vanishing of Ext over formal triangular matrix rings ［J］. Forum Mathematicum， 2006， 18（6）： 951-966.
HAGHANY A，MAZROOEI M，VEDADI M R. Pure projectivity and pure injectivity over formal triangular matrix rings ［J］. Journal of Algebra and Its Applications， 2012， 11（6）： 1250107.
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