MMN-1686

We study ideals that resemble $z$-ideals in commutative rings with identity. For each positive integer $n$, we say an ideal of a commutative ring $A$ is a $z^n$-ideal in case it has the property that if $a$ and $b$ belong to the same maximal ideals of $A$, and $a^n\in I$, then $b^n$ is also in $I$. The set of all $z^n$-ideals of $A$ is denoted by $\mathfrak{Z}^n(A)$. This gives an ascending chain $\mathfrak{Z}(A)\subseteq\mathfrak{Z}^2(A)\subseteq \mathfrak{Z}^3(A)\subseteq \cdots$ of collections of ideals, starting with the collection of $z$-ideals. We give examples of when the chain becomes stationary, and when it ascends without stop, with each collection properly contained in its successor. The assignment $A\mapsto \mathfrak{Z}^n(A)$ is shown to be the object part of a functor $\mathsf{Rng}_{\mathfrak{z}}^{\text{op}}\to\mathsf{Set}$, where $\mathsf{Rng}_{\mathfrak{z}}$ denotes the category of commutative rings with ring homomorphisms that contract $z$-ideals to $z$-ideals. When the objects are restricted to rings with zero Jacobson radical, the restricted functor reflects epimorphisms, but not monomorphisms.

Vol. 17 (2016), No. 1, pp. 171-185

DOI: 10.18514/MMN.2016.1686

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