ZWGP-injectivity and nonsingularity of rings
- LU Qi, BAO Hong-wei
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2017, 52(2):
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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.