On nonnil-pure theories

The notion of purity is fundamental in the theory of abelian groups. Zhao introduced the concepts of nonnil relatively divisible submodules and nonnil-pure submodules. In this paper, we explore additional properties of nonnil-pure submodules by defining what we term ‘nonnil-pure exact sequences’. This new definition extends the classical concept of purity originated by P. M. Cohn. Furthermore, inspired by Fieldhouse’s work, we extend the \(\phi \) $\varphi$ -von Neumann regular rings, previously introduced and studied by Tang, Wang, and Zhao, to modules.