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

Key words:

• 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