Reticulation of an integral complete l-groupoid: An axiomatic approach

The reticulation of a unital ring R is a bounded distributive lattice L(A) whose main property is that the Zariski prime spectrum of R is homeomorphic with the Stone prime spectrum of L(A). In this paper we develop an axiomatic theory of reticulation in the abstract framework offered by the integral complete l-groupoids (= icl-groupoids). For an algebraic icl-groupoid A we define axiomatically three basic notions: pre-reticulation, semi-reticulation and reticulation. We prove a uniqueness theorem for the semireticulations of A and we characterize the algebraic icl-groupoids that admit a reticulation. Given an algebraic icl-groupoid A, we define the spectral closure of the m-prime spectrum SpecZ(A) of A, an abstraction of the spectral closure of the prime spectrum of a ring. When A is the icl-groupoid Id(R) of ideals in a ring R we obtain some of the Belluce results on the reticulation of R and the spectral closure of the prime spectrum of R. (c) 2023 Elsevier B.V. All rights reserved.