Trivial extensions subject to semi-regularity and semi-coherence

Abstract

In this paper, we investigate the transfer of Matlis' semi-regularity and semi-coherence in trivial ring extensions issued from rings (with zero-divisors). We use the obtained results to enrich the literature with new examples of semi-regular or semi-coherent rings issued as trivial extensions and validate some questions left open in the literature.

Mathematics Subject Classification (2010): 13C10, 13C11, 13E05, 13F05, 13H10, 16A30, 16A50, 16A52.

Keywords: Trivial ring extension, idealization, semi-regular ring, IF ring, coherent ring, semi-coherent ring, arithmetical ring, chained ring, quasi-Frobenius ring, Von Neumann regular ring