Extensions of Injectivity and Coherent Rings

ABSTRACT

For a ring D and a subring C, let ℛ[D,C ] = {(d ₁,…, d _n , c,c,…): d _i  ∈ D, c ∈ C, n ≥ 1}, a subring of , and let , a subring of 𝕄₂(D). In this article, various conditions are obtained for ℛ[D,C ] and D∝ D to be right (m,n)-injective (in particular, P-injective, f-injective, FP-injective), right coherent, and right FC. Right nonsingular right (m,n)-injective rings are characterized using their maximal right quotient rings.

Communicated by I. Swanson.

Keywords:

• Annihilator
• Coherent ring
• Injectivity
• Trivial extension

Mathematics Subject Classification:

• Primary 16D50, 16S99

ACKNOWLEDGMENTS

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 was supported by NSERC (Grant OGP0194196), Canada.