On \phi-n-absorbing primary ideals of commutative rings

TL;DR

This paper introduces and studies the concept of ta-n-absorbing primary ideals in commutative rings, generalizing primary ideals through a function ta and an integer n, exploring their properties and structure.

Contribution

It defines ta-n-absorbing primary ideals, extending the theory of primary ideals with a new generalized framework and analyzing their fundamental properties.

Findings

Characterization of ta-n-absorbing primary ideals

Conditions for their existence and uniqueness

Relationships with classical primary ideals

Abstract

All rings are commutative with $1$ and $n$ is a positive integer. Let $\phi: J(R)\to J(R)\cup{\emptyset}$ be a function where $J(R)$ denotes the set of all ideals of $R$. We say that a proper ideal $I$ of $R$ is $\phi$-$n$-absorbing primary if whenever $a_1,a_2,...,a_{n+1}\in R$ and $a_1a_2\cdots a_{n+1}\in I\backslash\phi(I)$, either $a_1a_2\cdots a_n\in I$ or the product of $a_{n+1}$ with $(n-1)$ of $a_1,...,a_n$ is in $\sqrt{I}$. The aim of this paper is to investigate the concept of $\phi$-$n$-absorbing primary ideals.