On nonnil-finite condcutor rings

Adam Anebri; Najib Mahdou; El Houssaine  Oubouhou

doi:10.5269/bspm.70054

Adam Anebri Digital Engineering and Artificial Intelligence School, EuroMed University of Fez
Najib Mahdou
El Houssaine Oubouhou

Abstract

Let R be a commutative ring with nonzero identity and let $\mathcal{H}= \{R\mid R\text{ is a commutative ring and } Nil(R) \text{ is a divided prime ideal}\\ \text{of } R\}$. If $R\in \mathcal{H}$, then $R$ is called a $\phi$-ring. In this paper, we introduce and investigate new generalizations of nonnil-coherent rings. we introduce and study finite conductor and quasi coherent rings. $R$ is said to be a nonnil-finite conductor ring if $Ra\cap Rb$ and $(0:c)$ are finitely generated ideals of $R$ for all non-nilpotent elements $a,b,c\in R.$ $R$ is said to be a nonnil-quasi coherent ring if $a_1R \cap \cdots \cap a_nR$ and $(0 : c)$ are finitely generated ideals of $R$ for any finite set of non-nilpotent elements $c$ and $a_1, \ldots , a_n$ of $R$. Some basic properties of nonnil-finite conductor $($resp., nonnil-quasi coherent$)$ rings are studied. Further, we study the possible transfer to trivial ring extension and amalgamated algebra along an ideal. Examples illustrating the aims and scopes of our results are given.

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