JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2017, Vol. 52 ›› Issue (2): 19-23.doi: 10.6040/j.issn.1671-9352.0.2016.173

Previous Articles     Next Articles

ZWGP-injectivity and nonsingularity of rings

LU Qi, BAO Hong-wei   

  1. Department of Mathematics and Physics, Bengbu University, Bengbu 233030, Anhui, China
  • Received:2016-04-21 Online:2017-02-20 Published:2017-01-18

PDF (PC)

492

Abstract: ZWGP-injective modules and ZWGP-injective rings are introduced; equivalent characterizations of ZWGP-injective rings are obtained. The properties of ZWGP-injective modules(rings)are explored, the relations between ZWGP-injective rings and nonsingular rings are illustrated by some examples, and sufficient and necessary conditions that rings are nonsingular are given. It is proved that:(1)If left annihilators of R are W-ideals of R, and any essential ideals of R are left ZWGP-injective, then R is left nonsingular;(2)If for any essential left ideal I,R/I is ZWGP-injective, and l(a1)⊆l(a1a2)⊆l(a1a2a3)⊆… is stable, ai∈Z(RR), i=1,2,3,…, then R is left nonsingular.

Key words: WGP-injective ring, ZWGP-injective module, ZWGP-injective ring, GP-injective ring, nonsingular ring

CLC Number: 

  • O153.3
[1] ZHU Zhanmin. Almost JGP-injective rings and semipritive rings[J]. Procedia Engineering, 2012, 29:1697-1701.
[2] ZHAO Yu-e, DU Xianneng. Generalizations of Nil-injective rings[J]. Journal of Mathematical Research with Applications, 2014, 34(3):278-288.
[3] CAMILLO V, IBRAHIM Y, YOUSIF M F, et al. Simple-direct-injective modules[J]. Journal of Algebra, 2014, 420:39-53.
[4] 佟文廷. 同调代数引论[M]. 北京: 高等教育出版社, 1998:127-129. TONG Wenting. An introduction to homological algebra[M]. Beijing: Higher Education Press, 1998:127-129.
[5] NICHOLSON W K, YOUSIF M F. Principally injective rings[J]. Journal of Algebra, 1995, 174:77-93.
[6] CHEN Jianlong, DING Nanqing. On general principally injective rings[J]. Communications in Algebra, 1999, 27(5):2097-2116.
[7] LIANG Li. On a generalization of GP-injective rings[J]. Journal of Sichuan University: Natural Science, 2006, 43(4):726-729.
[8] GOODEARL K R. Ring theory, nonsingular rings and modules[M]. New York: Marcel Dekker, 1976:159-180.
[9] BEIDAR K I, JAIN S K, KANWAR P. Nonsingular CS-rings coincide with tight PP rings[J]. Journal of Algebra, 2004, 282:626-637.
[10] 鲁琦, 鲍宏伟,梁倞.环的WGP-内射性[J]. 山东大学学报(理学版), 2014,49(6): 35-39. LU Qi, BAO Hongwei, LIANG Liang. WGP-injectivity of rings[J]. Journal of Shandong University(Natural Science), 2014, 49(6): 35-39.
[11] ANDERSON F W, FULLER K R. Rings and categories of modules[M]. New York: Springer-Verlag, 1974:65-69.
[12] 周海燕. SF-环与正则环[D]. 芜湖:安徽师范大学, 2002. ZHOU Haiyan. SF-rings and regular rings[D]. Wuhu: Anhui Normal University, 2002.
[13] GOODEARL K R. Von Neumann regular rings[M]. Florida: Krieger Publishing Company, 1991:2-21.
[1] LU Qi, YIN Xiao-bin, BAO Hong-wei. EP-injectivity and von Neumann regularity of rings [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(10): 33-37.
[2] LU Qi1, BAO Hong-wei1, LIANG Liang2. WGP-injectivity of rings#br# [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(06): 35-39.
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