2022/05/23 by Cristina Flaut, Flaut, Cristina, Dana Piciu +1
Computer Science · Mathematics · #Advanced Algebra and Logic #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2205.11431
openalex publication_date 2022/05/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper using the connections between some subvarieties of residuated lattices, we investigated some properties of the lattice of ideals in commutative and unitary rings. We give new characterizations for commutative rings A in which Id(A) is an MV-algebra, a Heyting algebra or a Boolean algebra and we establish connections between these types of rings. We are very interested in the finite case and we present summarizing statistics. We show that the lattice of ideals in a finite commutative ring of the form A=% ℤ_k1× ℤ_k2× ...× ℤ% _kr, where ki=pi^αi and pi a prime number, for all i∈ \1,2,...,r\, is a Boolean algebra or an MV-algebra (which is not Boolean). Using this result we generate the binary block codes associated to the lattice of ideals in finite commutative rings and we present a new way to generate all (up to an isomorphism) finite MV-algebras using rings.