Reticulation of Quasi-commutative Algebras

Document Type : Research Paper

Author

Faculty of Mathematics, Bucharest University, Bucharest, Romania

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. Mure\c{s}an 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.

Keywords

References

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  • Receive Date: 24 August 2022
  • Accept Date: 26 August 2022

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