Zip and weak zip algebras in a congruence-modular variety

The zip (commutative) rings, introduced by Faith and Zelmanowitz, generated a fruitful line of investigation in ring theory. Recently, Dube, Blose and Taherifar developed an abstract theory of zippedness by means of frames. Starting from some ideas contained in their papers, we define and study the zip and weak zip algebras in a semidegenerate congruence-modular variety $\mathcal{V}$. We obtain generalizations of some results existing in the literature of zip rings and zipped frames. For example, we prove that a neo-commutative algebra $A\in \mathcal{V}$ is a weak zip algebra if and only if the frame $RCon(A)$ of radical congruences of $A$ is a zipped frame (in the sense of Dube and Blose). We study the way in which the reticulation functor preserves the  zippedness property. Using the reticulation and a Hochster’s theorem we prove that  a neo-commutative algebra $A\in \mathcal{V}$ is a weak zip algebra if and only if the minimal prime spectrum $Min(A)$ of $A$ is a finite space.