On regularly coherent modules and regularly Noetherian modules

The concepts of regular Noetherianity and regular coherence, which extend the classical notions of Noetherian and coherent rings, have been fundamental in the study of algebraic structures. In this paper, we aim to expand these notions to the realm of module theory. Specifically, we introduce and explore weak versions of injective, flat, and projective modules, which we term as reg-injective, reg-flat, and reg-projective modules. We provide analogues of classical results and establish their properties, offering examples to illustrate modules that meet these new criteria but not their classical counterparts. Additionally, we define and study regularly Noetherian and regularly coherent modules, characterizing their properties and examining their stability under various ring constructions. Our results contribute new examples and broaden the current understanding of these algebraic concepts.