Czechoslovak Mathematical Journal, Vol. 75, No. 3, pp. 1073-1091, 2025
$(m,n)$-prime ideals of commutative rings
Hani A. Khashan, Ece Yetkin Çelikel
Received March 17, 2025. Published online August 18, 2025.
Abstract: Let $R$ be a commutative ring with identity and $m$, $n$ be positive integers. 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 $a,b\in R$, $a^mb\in I$ implies either $a^n\in I$ or $b\in I.$ Several characterizations of this new class with many examples are given. Analogous 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.
Affiliations: Hani A. Khashan, Department of Mathematics, Faculty of Science, Al al-Bayt University, P.O.BOX 130040, Al Mafraq 25113, Jordan, e-mail: hakhashan@aabu.edu.jo; Ece Yetkin Çelikel (corresponding author), Department of Basic Sciences, Faculty of Engineering, Hasan Kalyoncu University, Gaziantep, Turkey, e-mail: ece.celikel@hku.edu.tr, yetkinece@gmail.com
