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- Title
RETICULATION OF QUASI-COMMUTATIVE ALGEBRAS.
- Authors
GEORGESCU, G.
- Abstract
The commutator theory, developed by Fresee and McKenzie in the framework of 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. Muresan 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.
- Subjects
COMMUTATIVE algebra; COMMUTATORS (Operator theory); GEOMETRIC congruences; HOMEOMORPHISMS; NONCOMMUTATIVE rings
- Publication
Journal of Mahani Mathematical Research Center, 2023, Vol 12, Issue 2, p115
- ISSN
2251-7952
- Publication type
Article
- DOI
10.22103/jmmr.2022.20133.1328