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.