Matlis’ semi-regularity and semi-coherence in trivial ring extensions: a survey

In 1985, Matlis defined a ring R to be semi-coherent if HomR(MN) is a submodule of a flat R-module for any injective R-modules M and N. Then, inspired by this definition and von Neumann regularity, he defined a ring to be semi regular if any module can be embedded in a flat module. He then established a connection of this notion with coherence by proving that semi-regularity is a local property in the class of coherent rings. This survey paper tracks three recent works which investigate these notions in trivial ring extensions (also called Nagata idealizations) and contribute to the study of homological aspects of these constructions. Section 2 deals with the transfer of semi-regularity along with related concepts, such as coherence, in trivial ring extensions issued from integral domains. All along the section, the new results are put in use to enrich the literature with new families of examples subject to semi-regularity. Section 3 features the transfer of semi-regularity and semi-coherence in trivial ring extensions issued from rings (with zero-divisors). The new results are used to enrich the literature with new examples of semi-regular or semi-coherent rings which arise as trivial ring extensions, and hence validate some questions left open in the literature. Section 4 proves an extension of Zaks’ conjecture on integral domains with semi-regular proper homomorphic images (with respect to finitely generated ideals) to arbitrary rings (i.e., possibly with zero-divisors). The main result extends and recovers two classical results, due to Levy (1966) and Matlis (1985), on Noetherian rings and Prüfer domains, respectively. It also proves a global version of a result of Couchot (2003) on chained (valuation) rings. As an application, new examples of rings with semi-regular proper homomorphic images are constructed through transfer results on trivial ring extensions.