三角矩阵环上的absolutely clean模与GorensteinAC-平坦模

doi:10.6040/j.issn.1671-9352.0.2020.200

《山东大学学报(理学版)》 ›› 2020, Vol. 55 ›› Issue (12): 81-88.doi: 10.6040/j.issn.1671-9352.0.2020.200

• • 上一篇

三角矩阵环上的absolutely clean模与GorensteinAC-平坦模

狄振兴,李晓曼^*

西北师范大学数学与统计学院, 甘肃兰州 730070

发布日期:2020-12-01
作者简介:狄振兴(1984— ), 男, 副教授, 博士研究生, 硕士生导师, 研究方向为同调代数理论. E-mail:dizhenxing19841111@126.com*通信作者简介:李晓曼(1998— ), 男, 硕士研究生, 研究方向为同调代数理论. E-mail:2407663174@qq.com
基金资助:
国家自然科学基金资助项目(11601433,11971388)

Absolutely clean modules and Gorenstein AC flat modules over formal triangular matrix rings

DI Zhen-xing, LI Xiao-man^*

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China

Published:2020-12-01

摘要/Abstract

摘要： 设T=(A 0U B),其中A,B是环,U是左B右A双模。研究了absolutely clean T-模与Gorenstein AC-平坦T-模。在一定条件下,证明了L=(L₁,L₂)是absolutely clean右T-模,则L₁是absolutely clean右A-模且L₂是absolutely clean右B-模。作为这个结果的应用,在一定条件下,进一步证明了M=(M₁M₂)_φ^M是Gorenstein AC-平坦左T-模当且仅当M₁是Gorenstein AC-平坦左A-模,M₂/Im φ^M是Gorenstein AC-平坦左B-模并且φ^M是单同态。

关键词: absolutely clean模, Gorenstein AC-平坦模, 三角矩阵环

Abstract: Let T=(A 0U B) be a formal triangular matrix ring, where A and B are rings and U is a (B,A)-bimodule. In this article, we investigate absolutely clean T-modules and Gorenstein AC-flat T-modules. More specifically, under some mild conditions, we prove that L=(L1, L2) is absolutely clean right T-module then L1 is absolutely clean right A-module and L₂ is absolutely clean right B-module. As an application of this result, under some conditions, we show that M=(M1M2)_φ^M is Gorenstein AC flat left T-module if and only if M1 is a Gorenstein AC flat left A-module, M2/Im φ^M is a Gorenstein AC flat left B-module and the morphism φ^M is a monomorphism.

Key words: absolutely clean module, Gorenstein AC flat module, formal triangular matrix ring

中图分类号:

O153.3

狄振兴,李晓曼. 三角矩阵环上的absolutely clean模与GorensteinAC-平坦模[J]. 《山东大学学报(理学版)》, 2020, 55(12): 81-88.

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.

参考文献

[1] FOSSUM R M, GRIFFITH P A, REITEN I. Trivial extensions of Abelian categories[M] //Homological Algebra of Trivial Extensions of Abelian Catergories with Applications to Ring Theory. Berlin: Springer, 1975.
[2] GOODEARL K R, Jr WARFIELD R B. An introduction to noncommutative Noetherian rings[M]. Cambridge: Cambridge University Press, 2004.
[3] HAGHANY A, VARADARAJAN K. Study of modules over formal triangular matrix rings[J]. Journal of Pure and Applied Algebra, 2000, 147(1):41-58.
[4] BRAVO D, GILLESPIE J, HOVEY M. The stable module category of a general ring[J/OL]. 2014. https://ui.adsabs.harvard.edu/abs/2014arXiv1405.5768B/abstract
[5] BRAVO D, ESTRADA S, IACOB A. FP n-injective, FP n-flat covers and preenvelopes, and Gorenstein AC-flat covers[C] //Algebra Colloquium. Singapore: World Scientific Publishing Company, 2018, 25(2):319-334.
[6] GREEN E L. On the representation theory of rings in matrix form[J]. Pacific Journal of Mathematics, 1982, 100:123-138.
[7] MAO Lixin. Cotorsion pairs and approximation classes over formal triangular matrix rings[J]. Journal of Pure and Applied Algebra, 2020, 224(6):106-135.
[8] ROTMAN J J. An introduction to homological algebra[M]. Berlin: Academic, 2009.
[9] ESTRADA S, IACOB A, PÉREZ M A. Model structures and relative Gorenstein flat modules and chain complexes[J/OL]. 2017. https://xueshu.baidu.com/usercenter/paper/show?paperid=e0bab0c50d6d83a72db5fc1cada7ab23&site=xueshu_se&hitarticle=1

多维度评价

Viewed

Full text

Abstract

Cited

Shared

Discussed

三角矩阵环上的absolutely clean模与GorensteinAC-平坦模

Absolutely clean modules and Gorenstein AC flat modules over formal triangular matrix rings

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 8

多维度评价

本文评价

推荐阅读 0

[1]	杨丽娜,王尧,任艳丽. 斜三角矩阵环的几个新结果[J]. 《山东大学学报(理学版)》, 2020, 55(12): 49-55.
[2]	赵阳,张文汇. 形式三角矩阵环上的强Ding投射模和强Ding内射模[J]. 《山东大学学报(理学版)》, 2020, 55(10): 37-45.
[3]	马广琳,王尧,任艳丽. T-幂零环的若干扩张[J]. 《山东大学学报(理学版)》, 2019, 54(12): 68-73.
[4]	朱荣民, 王占平. n阶三角矩阵环上的 Gorenstein 投射模与维数[J]. 山东大学学报（理学版）, 2015, 50(12): 85-92.
[5]	黄述亮. 形式三角矩阵环上导子的几个结果[J]. 山东大学学报（理学版）, 2015, 50(10): 43-46.
[6]	王文康. 中心线性 McCoy 环[J]. J4, 2013, 48(12): 6-13.
[7]	王文康. 关于弱斜Armendariz环[J]. J4, 2010, 45(10): 15-19.
[8]	王文康 . 一类上三角矩阵环的Armendariz与半交换性质[J]. J4, 2008, 43(2): 62-65 .