Journal of Guangxi Normal University(Natural Science Edition) ›› 2019, Vol. 37 ›› Issue (2): 113-120.doi: 10.16088/j.issn.1001-6600.2019.02.013

Previous Articles     Next Articles

Graded Simple Injective Modules and Their Characterization

LIANG Chunmei, WANG Fanggui*, WU Xiaoying   

  1. College of Mathematics and Software Science, Sichuan Normal University, Chengdu Sichuan 610068, China
  • Received:2018-04-02 Online:2019-04-25 Published:2019-04-28

Abstract: In this paper, the concept of graded simple injective modules is introduced. Let R be a G-graded ring. A graded R-module N is called a graded single injective module if EXT1R(S,N)=0 for any graded simple R-module S. A series of equivalent characterizations for graded simple injective modules are given, and it is shown that: if R is a left graded Artinian ring or R is a graded Noetherian ring with the graded Krull dimension at most one, then a graded module E is graded injective if and only if E is graded simple injective.

Key words: graded rings, graded simple modules, graded injective modules, graded simple injective modules, graded Noetherian rings

CLC Number: 

  • O153.3
[1] WANG Mingyi, ZHAO Guo. On maximal injectivity[J]. Acta Mathematical Sinica, 2005, 21(1):1451-1458.DOI:10.1007/s10114-005-0599-0.
[2] 王芳贵, 汪明义, 杨立英. 交换环上的极大性内射模[J]. 四川师范大学学报(自然科学版), 2010, 33(3):277-285.DOI:10.3969/j.issn.1001-8395.2010.03.001.
[3] 王修建, 杜先能. 极大内射性与极大平坦性的一些性质[J]. 吉林大学学报(理学版), 2012, 50(4):628-632.DOI:10.13413/j.cnki.jdxblxb.2012.04.034.
[4] 邹红林, 陈淼森. 关于单投射模[J]. 浙江师范大学学报(自然科学版), 2007, 30(4):410-415.DOI:10.3969/j.issn.1001-5051.2007.04.010.
[5] 梁莉莉, 王芳贵,熊涛. 强单投射模与强单内射模[J]. 黑龙江大学自然科学学报, 2016, 33(6):728-734.DOI:10.13482/j.issn.1001-7011.2016.04.217.
[6] N$\check{A}$ST$\check{A}$SESCU C, OYSTAEYEN F V. Methods of graded rings[M]. New York:Springer, 2004.
[7] N$\check{A}$ST$\check{A}$SESCU C, OYSTAEYEN F V. Graded ring theory[M]. New York:North-Holland Publishing Co., 1982.
[8] CHEN T S, HUANG C F, LIANG J W. Extended Jacobson density theorem for graded rings with derivations and automorphisms[J]. Taiwanese Journal of Mathematics, 2010, 14(5):1993-2014.DOI:10.11650/twjm/1500406028.
[9] YANG Xiaoyan, LIU Zhongkui. FP-gr-injective modules[J]. Mathematical Journal of Okayama University, 2011, 53:83-100.DOI:10.1201/9780203903889.ch1.
[10] 张积成. Gr-弱半局部环的同调性质[J]. 广西师范大学学报(自然科学版), 1998, 16(1):20-24.DOI:10.16088/j.issn.1001-6600.1998.01.004.
[11] 赵巨涛, 张积成. 分次环的分次Jacobson根-分次局部环和分次Artin环[J]. 长治学院学报, 2001, 18(3):1-4.DOI:10.3969/j.issn.1673-2014.2001.03.001.
[12] 赖章荣. Noether 分次环上分次素理想的若干性质[J]. 赣南师范学院学报, 2000(6):19-21.DOI:10.13698/j.cnki.cn36-1037/c.2000.06.007.
[13] PARK C H, PARK M H. Integral closure of a graded Noetherian domain[J]. Journal of the Korean Mathematical Society, 2011, 48(3):449-464.DOI:10.4134/jkms.2011.48.3.449.
[14] 王刚. 分次Dedekind 整环[D]. 成都:四川师范大学, 2013.
[1] WEI Yangjiang, LIANG Yiyao, TANG Gaohua, SU Leilei, CHEN Weining. Cubic Mapping Graphs on the Quotient Ringsof the Gaussian Integer Rings of Modulo n [J]. Journal of Guangxi Normal University(Natural Science Edition), 2016, 34(3): 53-61.
[2] TANG Gaohua,LI Yu, WU Yansheng. Properties of Zero-divisor Graph of Group Ring Zn[i]G [J]. Journal of Guangxi Normal University(Natural Science Edition), 2016, 34(4): 109-115.
[3] GUO Shu-feng, XIE Guang-ming, YI Zhong. Properties of Zero-divisor Graphs of Group Rings ZnG [J]. Journal of Guangxi Normal University(Natural Science Edition), 2015, 33(2): 68-75.
[4] WEI Yin-hu, PANG Gui-xi, WU Jiao, XIE Guang-ming. Invariant Gauss Extensions in Q(K G) [J]. Journal of Guangxi Normal University(Natural Science Edition), 2013, 31(2): 55-57.
[5] HUANG Yi-fei, YI Zhong, QIN Qing-ling. Zero-divisor Graph of Group Ring ZnD4 [J]. Journal of Guangxi Normal University(Natural Science Edition), 2011, 29(2): 15-20.
[6] GUO Shufeng , XIE Guangming, YI Zhong. Properties of Zero-divisor Graphs of Idealizations of Group Rings ZnG [J]. Journal of Guangxi Normal University(Natural Science Edition), 2016, 34(1): 66-71.
[7] TANG Gaohua, WU Yansheng, ZHANG Hengbin, LI Yu. Strong 2-sum Property of Local Rings and Their Extensions [J]. Journal of Guangxi Normal University(Natural Science Edition), 2016, 34(1): 72-77.
Viewed
Full text


Abstract

Cited

  Shared   
  Discussed   
No Suggested Reading articles found!