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

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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

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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
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