JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2022, Vol. 57 ›› Issue (2): 38-44.doi: 10.6040/j.issn.1671-9352.0.2021.328

Previous Articles    

Gorenstein FP-injective modules over formal triangular matrix rings

YANG Yin-yin, ZHANG Cui-ping*   

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China
  • Published:2022-01-07

PDF (PC)

389

Abstract: Let T=(A 0U B) be a formal triangular matrix ring, where A and B are rings and U is (B,A)-bimodule. It is proved that if T is a left cocherent ring, BU is finitely presented and pd(BU)<∞, M=(M1M2)φM is Gorenstein FP-injective left T-module, then Ker φM^~ is Gorenstein FP-injective left A-module, M2 is Gorenstein FP-injective left B-module, and φM^~ is an epimorphism; if T is still a left GFPI-closed ring, then the opposite case holds.

Key words: formal triangular matrix ring, FP-injective module, Gorenstein FP-injective module

CLC Number: 

  • O153.3
[1] HAGHANY A, VARADARAJAN K. Study of formal triangular matrix rings[J]. Comm Algebra, 1999, 27(11):5507-5525.
[2] HAGHANY A, VARADARAJAN K. Study of modules over formal triangular matrix rings[J]. J Pure Appl Algebra, 2000, 147:41-58.
[3] ENOCHS E E, JENDA O M G. Relative homological algebra[M]. Berlin: Walter de Gruyter, 2000.
[4] MAO Lixin, DING Nanqing. Gorenstein FP-injective and Gorenstein flat modules[J]. Journal of Algebra and Its Application, 2008, 7(4):491-506.
[5] GAO Zenghui, WANG Fanggui. Coherent rings and Gorenstein FP-injective modules[J]. Comm Algebra, 2012, 40(5):1669-1679.
[6] ZHANG Pu. Gorenstein projective modules and symmetric recollements[J]. J Algebra, 2013, 388:65-80.
[7] ENOCHS E E, CORTES-IZURDIAGA M. Gorenstein conditions over triangular matrix rings[J]. J Pure Appl Algebra, 2014, 218:1544-1554.
[8] MAO Lixin. Gorenstein flat modules dimension over formal triangular matrix rings[J]. J Pure Appl Algebra, 2019, 224:1-10.
[9] MAO Lixin. Duality pairs and FP-injective modules over formal triangular matrix rings[J]. Comm Algebra, 2020, 48:5296-5310.
[10] 杨燕妮,杨刚.关于Gorenstein FP-内射模及维数[J]. 四川师范大学学报(自然科学版),2016, 39(1):47-50. YANG Yanni, YANG Gang. On Gorenstein FP-injective modules and dimensions[J]. Journal of Sichuan Normal University(Natural Science), 2016, 39(1):47-50.
[11] 夏国利,王芳贵. 形式三角矩阵环上的PC-内射模[J]. 四川师范大学学报(自然科学版),2018, 41(1):39-43. XIA Guoli, WANG Fanggui. PC-injective modules over formal triangular matrix rings[J]. Journal of Sichuan Normal University(Natural Science), 2018, 41(1):39-43.
[12] 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):1-13.
[1] WU De-jun, ZHOU Hui. A characterization of Gorenstein FP-injective modules over triangular matrix rings [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(8): 86-93.
[2] LIU Ting, ZHANG Wen-hui. Gorenstein IFP-Flat modules [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(2): 91-98.
[3] DI Zhen-xing, LI Xiao-man. Absolutely clean modules and Gorenstein AC flat modules over formal triangular matrix rings [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(12): 81-88.
[4] ZHAO Yang, ZHANG Wen-hui. Strongly Ding projective and strongly Ding injective modules over formal triangular matrix rings [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(10): 37-45.
[5] WU Xiao-ying, WANG Fang-gui. Graded version of Enochs theorem [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(10): 22-26.
[6] HUANG Shu-liang. Several results on derivations of formal triangular matrix rings [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(10): 43-46.
[7] CHEN Wen-jing, LIU Zhong-kui. Remarks on n-strongly Gorenstein FP-injective modules [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(04): 50-54.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd