(m,n)-CLOSED AND QUASI (m,n)-CLOSED IDEALS ON COMMUTATIVE RINGS WITH IDENTITY

https://doi.org/10.22146/jmt.106690

Erlangga Adinugroho Rohi(1*), Sri Wahyuni(2)

(1) Universitas Gadjah Mada
(2) Universitas Gadjah Mada
(*) Corresponding Author

Abstract


Throughout this paper, all rings considered are commutative rings $R$ with identity $1_R$. Let $m$ and $n$ be natural numbers such that $1\leq n<m$. A proper ideal $I$ of $R$ is called an $(m,n)-$closed ideal if for every $x\in R$ with $x^m\in I$ implies $x^n\in I$. An $(m,n)-$closed ideal generalizes semi $n-$absorbing ideal and, hence, also generalizes semiprime ideal. A proper ideal $I$ of $R$ is called a quasi $(m,n)-$closed ideal if for every $x\in R$ with $x^m\in I$ implies $x^{n}\in I$ or $x^{m-n}\in I$. Therefore, a quasi $(m,n)-$closed generalizes an $(m,n)-$closed ideal. Research related to these ideals is referred to Anderson and Badawi (2017) and Khashan and Celikel (2024). In this paper, the authors presented several new properties related to these ideals that are not discussed in the two main references.

Keywords


(m,n)-closed ideal; quasi (m,n)-closed ideal; semiprime ideal; commutative ring with identity

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DOI: https://doi.org/10.22146/jmt.106690

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