*On ϕ-(n,J)-ideals and n-J-ideals of commutative rings*

**Abdelhaq El Khalfi **

Fundamental and Applied Mathematics Laboratory

Faculty of Sciences Ain Chock, Hassan II University of Casablanca, Morocco.

Pages 143-153 | Received 10 December 2022, Accepted 04 April 2023, Published 16 June 2023

**Abstract**

Let R be a commutative ring with nonzero identity. In this paper, we introduce and investigate a generalization of (2,J)-ideals. Let \phi : \mathcal{I}(R)\rightarrow \mathcal{I} (R)\cup {\emptyset} be a function where \mathcal{I}(R) is the set of ideals of R. A proper ideal I of R is said to be a \phi–(n,J)-ideal if whenever x_1\cdots x_{n+1}\in I\setminus \phi(I), for x_1,\ldots,x_{n+1}\in R, then x_1\cdots x_n\in I or x_1\cdots \widehat{x_k}\cdots x_{n+1}\in Jac(R), for some k\in {1,\ldots,n}. Also, I is called an n–J-ideal if whenever x_1\cdots x_{n+1}\in I, for x_1,\ldots,x_{n+1}\in R, then x_1\cdots x_n\in Jac(R) or x_1\cdots \widehat{x_k}\cdots x_{n+1}\in I, for some k\in {1,\ldots,n}. Moreover, we give some basic properties of those classes of ideals and we study the \phi–(n,J)-ideals and the n–J-ideals of the localization of rings, the direct product of rings, the trivial ring extensions and the amalgamation of rings.

**Keywords**: \phi–(n,J)-ideal, \phi–n-absorbing primary ideal, \phi–n-absorbing ideal, \phi-prime ideal, n–J-ideal.

**MSC numbers**: Primary 13A15; Secondary 13C05.

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