On weakly $(m,n)$-prime ideals of commutative rings
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Let $R$ be a commutative ring with identity and $m$, $n$ be positive integers. In this paper, we introduce the class of weakly $(m,n)$-prime ideals generalizing $(m,n)$-prime and weakly $(m,n)$-closed ideals. A proper ideal $I$ of $R$ is called weakly $(m,n)$-prime if for $a,b\in R$, $0\neq a^{m}b\in I$ implies either $a^{n}\in I$ or $b\in I$. We justify several properties and characterizations of weakly $(m,n)$-prime ideals with many supporting examples. Furthermore, we investigate weakly $(m,n)$-prime ideals under various contexts of constructions such as direct products, localizations and homomorphic images. Finally, we discuss the behaviour of this class of ideals in idealization and amalgamated rings.
Abstract
Let $R$ be a commutative ring with identity and $m$, $n$ be positive integers. In this paper, we introduce the class of weakly $(m,n)$-prime ideals generalizing $(m,n)$-prime and weakly $(m,n)$-closed ideals. A proper ideal $I$ of $R$ is called weakly $(m,n)$-prime if for $a,b\in R$, $0\neq a^{m}b\in I$ implies either $a^{n}\in I$ or $b\in I$. We justify several properties and characterizations of weakly $(m,n)$-prime ideals with many supporting examples. Furthermore, we investigate weakly $(m,n)$-prime ideals under various contexts of constructions such as direct products, localizations and homomorphic images. Finally, we discuss the behaviour of this class of ideals in idealization and amalgamated rings.
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