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Keywords:
$(m,n)$-prime ideal; $(m,n)$-closed ideal; $n$-absorbing ideal; avoidance theorem
$(m,n)$-prime ideal; $(m,n)$-closed ideal; $n$-absorbing ideal; avoidance theorem
Summary:
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 $a,b\in R$, $a^{m}b\in I$ 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 \hbox {$(m,n)$-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.
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