New results on Congruence Boolean Lifting Property

Document Type : Original Article

Author

University of Bucharest, Faculty of Mathematics and Computer Science, Bucharest, Romania

Abstract

The Lifting Idempotent Property (LIP) of ideals in commutative rings inspired the study of Boolean lifting properties in the context of other concrete algebraic structures (MV-algebras, commutative l-groups, BL-algebras, bounded distributive lattices, residuated lattices, etc.), as well as for some types of universal algebras (C. Muresan and the author defined and studied the Congruence Boolean Lifting Property (CBLP) for congruence modular algebras). A lifting ideal of a ring R is an ideal of R fulfilling LIP. In a recent paper, Tarizadeh and Sharma obtained new results on lifting ideals in commutative rings. The aim of this paper is to extend an important part of their results to congruences with CBLP in semidegenerate congruence modular algebras. The reticulation of such algebra will play an important role in our investigations (recall that the reticulation of a congruence modular algebra A is a bounded distributive lattice L(A) whose prime spectrum is homeomorphic with Agliano's prime spectrum of A). Almost all results regarding CBLP are obtained in the setting of semidegenerate congruence modular algebras having the property that the reticulations preserve the Boolean center. The paper contains several properties of congruences with CBLP. Among the results we mention a characterization theorem of congruences with CBLP. We achieve various conditions that ensure CBLP. Our results can be applied to a lot of types of concrete structures: commutative rings, l-groups, distributive lattices, MV-algebras, BL-algebras, residuated lattices, etc.