reticulation of quasi-commutative algebras

The commutator operation in a congruence-modular variety v allows us to define the prime congruences of any algebra a ∈ 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 ∈ 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¸san and the author defined the reticulation for the algebras a in a semidegenerate congruence-modular variety 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 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.