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(m, n)-Prime Ideals of Commutative Rings
Version 1
: Received: 3 January 2024 / Approved: 5 January 2024 / Online: 5 January 2024 (14:21:25 CET)
How to cite: Khashan, H.; Yetkin Celikel, E. (m, n)-Prime Ideals of Commutative Rings. Preprints 2024, 2024010472. https://doi.org/10.20944/preprints202401.0472.v1 Khashan, H.; Yetkin Celikel, E. (m, n)-Prime Ideals of Commutative Rings. Preprints 2024, 2024010472. https://doi.org/10.20944/preprints202401.0472.v1
Abstract
Let R be a commutative ring with identity and m, n be positive integers. In this paper, we introduce the class of (m,n)-prime ideals which lies properly between the classes of prime and (m,n)-closed ideals. A proper ideal I of R is called (m,n)-prime if for a,b∈R, a^{m}b∈I implies either aⁿ∈I or b∈I. Several characterizations of this new class with many examples are given. Analougus to primary decomposition, we define the (m,n)-decomposition of ideals and show that every ideal in an n-Noetherian ring has an (m,n)-decomposition. Furthermore, the (m,n)-prime avoidance theorem is proved.
Keywords
(m; n)-prime ideal; (m; n)-closed ideal; n-absorbing ideal; avoidance theorem
Subject
Computer Science and Mathematics, Algebra and Number Theory
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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