On S-(m,n)-absorbing ideals and S-(m,n)-absorbing prime ideals of commutative rings
Mohammed Assalami\,^1, Ayoub Kharbouch\,^2 and Hwankoo Kim\,^3
\,^{1,2} Department of Mathematics, Faculty of Science and Technology of Fez, Box 2202, University S.M. Ben Abdellah Fez, Morocco.
\,^{3} Division of Computer Engineering, Hoseo University, Asan 31499, Republic of Korea.
Pages 43-59 | Received 05 April 2024, Accepted 24 June 2025, Published 19 July 2026
Abstract
Let R be a commutative ring with identity, S a multiplicative subset of R, and I a proper ideal of R disjoint from S. Let m and n be positive integers with m > n. In this paper, we introduce the concepts of S\text{-}(m,n)-absorbing ideals, S\text{-}(m,n)-absorbing prime ideals, and S\text{-}AB\text{-}(m,n)-absorbing ideals. These notions generalize several existing concepts, including (m,n)-absorbing ideals, (m,n)-absorbing prime ideals, AB\text{-}(m,n)-absorbing ideals, and S\text{-}n-absorbing ideals. We investigate the fundamental properties of these new classes of ideals and examine their behavior under various ring constructions, such as quotient rings, homomorphic images, and localizations. Furthermore, we explore the transfer of these properties to trivial ring extensions and amalgamated rings.
Keywords: S\text{-}(m,n)-absorbing ideal, (m,n)-absorbing ideal, S\text{-}n-absorbing ideal, S-prime ideal, trivial extension, amalgamated algebra.
MSC numbers: Primary 13A15; Secondary 13B99, 13B30.
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