GP-Injective Rings Need Not be P-Injective

Abstract

A ring R is called left P-injective if for every a ∈ R, aR = r(l(a)) where l( ⋅ ) and r( ⋅ ) denote left and right annihilators respectively. The ring R is called left GP-injective if for any 0 ≠ a ∈ R, there exists n > 0 such that a ⁿ  ≠ 0 and a ⁿ R = r(l(a ⁿ )). As a response to an open question on GP -injective rings, an example of a left GP-injective ring which is not left P-injective is given. It is also proved here that a ring R is left FP -injective if and only if every matrix ring 𝕄 _n (R) is left GP-injective.

Key Words:

• FP-injective ring
• GP-injective ring
• Matrix ring
• P-injective ring

Mathematics Subject Classification:

• Primary 16D50

ACKNOWLEDGMENTS

The research was carried out during a visit by the first author to Memorial University of Newfoundland. He would like to gratefully acknowledge the financial support and kind hospitality from his host institute. The first author was supported by the National Natural Science Foundation of China (No. 10171011) and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutes of MOE, P.R.C. The second author is supported by NSERC (Grant OGP0194196) and a grant from the Office of Dean of Science, Memorial University.

Notes

^#Communicated by R. Wisbauer.