Connections between commutative rings and some algebras of logic

In this paper using the connections between some subvarieties of residuatedlattices, we investigated some properties of the lattice of ideals incommutative and unitary rings. We give new characterizations for commutativerings A in which Id(A) is an MV-algebra, a Heyting algebra or a Booleanalgebra and we establish connections between these types of rings. We arevery interested in the finite case and we present summarizing statistics. Weshow that the lattice of ideals in a finite commutative ring is a Booleanalgebra or an MV-algebra (which is not Boolean).Using this result we generate the binary block codes associated to thelattice of ideals in finite commutative rings and we present a new way togenerate all (up to an isomorphism) finite MV-algebras using rings.

Keywords:

Commutative ring , Ideal , BCK-algebra , residuated lattice , MV-algebra , Boolean algebra , Heyting algebra , Chang property , block codes