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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 zn-ideal in case it has the property that if a and b belong to the same maximal ideals of A, and an ϵ I , then bn is also in I. The set of all zn-ideals of A is denoted by A --> ʒn (A). This gives an ascending chain ʒ(A) < ʒ2(A) < ʒ3(A) <… 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 à ʒn (A) is shown to be the object part of a functor Rngʒop --> Set, where Rngʒop 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.


This article was originally published in Miskolc Mathematical Notes, volume 17, issue 1, in 2016.

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Miskolc University Press

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Algebra Commons



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