# Uncountable families of prime z-ideals in C 0R - Mathematics > Rings and Algebras

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Abstract: Denote by $\continuum=2^{\aleph 0}$ the cardinal of continuum. We constructan intriguing family $P \alpha: \alpha\in\continuum$ of prime $z$-ideals in$\C 0 eals$ with the following properties:If $f\in P {i 0}$ for some $i 0\in\continuum$, then $f\in P i$ for all butfinitely many $i\in \continuum$;$\bigcap {i eq i 0} P i subset P {i 0}$ for each $\i 0\in \continuum$.We also construct a well-ordered increasing chain, as well as a well-ordereddecreasing chain, of order type $\kappa$ of prime $z$-ideals in $\C 0 eals$for any ordinal $\kappa$ of cardinality $\continuum$.

Author: ** Hung Le Pham**

Source: https://arxiv.org/