\phi-rings from a module-theoretic point of view: a survey
Hwankoo Kim\,^1, Najib Mahdou\,^2 , and El Houssaine Oubouhou\,^2
^1\,Division of Computer Engineering, Hoseo University, Asan 31499, Republic of Korea.
^2\,Laboratory of Modelling and Mathematical Structures,
Department of Mathematics, Faculty of Science and Technology of Fez, Box 2202, University S.M. Ben Abdellah Fez, Morocco.
Pages 78-114 | Received 02 September 2023, Accepted 13 October 2023, Published 30 June 2024
Abstract
Let R be a commutative ring with a nonzero identity and Nil(R) be its set of nilpotent elements. Recall that a prime ideal P of R is called divided prime if P \subset (x) for every x \in R\setminus P; thus a divided prime ideal is comparable to every ideal of R. In many articles, the authors investigated the class of rings \mathcal{H} = \{R\ |\ R \text{ is a commutative ring and } Nil(R) \text{ is a divided prime ideal of } R\} (observe that if R is an integral domain, then R \in \mathcal{H}). If R \in \mathcal{H}, then R is called a \phi-ring. In this paper, we survey known results concerning \phi-rings from a module-theoretic point of view.
Keywords: \phi-ring, \phi-Prüfer, nonnil-Noetherian, \phi-Dedekind, \phi-torsion, \phi-flat, nonnil-injective, \phi-von Neumann regular, nonnil-coherent, nonnil-commutative diagram, \phi-(weakly) global dimension, nonnil-projective.
MSC numbers: Primary 13A15, 13C11, 13D05; Secondary 13A15, 13B02, 3C05, 13C12, 13E15, 13F05.
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