The aim of this paper is to define an abstract quantale framework for extending some properties of the zip rings (studied by Faith, Zelmanowitz, etc.) and the weak zip rings (defined by Ouyang). By taking as prototype the quantale of ideals of a zip ring (resp. a weak zip ring) we introduce the notion of zipped quantale (resp. weakly zipped quantale). The zipped quantales also generalize the zipped frames, defined by Dube and Blose in a recent paper. We define the zip (bounded distributive) lattices and we prove that a coherent quantale A is weakly zipped iff the reticulation L(A) of A is a zip lattice. From this result we obtain the following corollary: the coherent quantale A is weakly zipped iff the frame R(A) of the radical elements of A is zipped. Such theorems allow us to extend to quantale framework a lot of results obtained by Dube and Blose for the zipped frames and for the weak zip rings.

The commutator theory, developed by Fresee and McKenzie in the framework of a congruence-modular variety $\mathcal{V}$, allows us to define the prime congruences of any algebra $A\in \mathcal{V}$ and the prime spectrum $Spec(A)$ of $A$. The first systematic study of this spectrum can be found in a paper by Agliano, published in Universal Algebra (1993).The reticulation of an algebra $A\in \mathcal{V}$ is a bounded distributive algebra $L(A)$, whose prime spectrum (endowed with the Stone topology) is homeomorphic to $Spec(A)$ (endowed with the topology defined by Agliano). In a recent paper, C. Mure\c{s}an and the author defined the reticulation for the algebras $A$ in a semidegenerate congruence-modular variety $\mathcal{V}$, satisfying the hypothesis $(H)$: the set $K(A)$ of compact congruences of $A$ is closed under commutators. This theory does not cover the Belluce reticulation for non-commutative rings. In this paper we shall introduce the quasi-commutative algebras in a semidegenerate congruence-modular variety $\mathcal{V}$ as a generalization of the Belluce quasi-commutative rings. We define and study a notion of reticulation for the quasi-commutative algebras such that the Belluce reticulation for the quasi-commutative rings can be obtained as a particular case. We prove a characterization theorem for the quasi-commutative algebras and some transfer properties by means of the reticulation.