TY - JOUR
ID - 7213
TI - Connections between commutative rings and some algebras of logic
JO - Iranian Journal of Fuzzy Systems
JA - IJFS
LA - en
SN - 1735-0654
AU - Flaut, C.
AU - Piciu, D.
AD - Faculty of Mathematics and Computer Science, Ovidius University, Bd. Mamaia 124, 900527, Constanta, Romania
AD - Faculty of Science, University of Craiova, A.I. Cuza Street, 13, 200585, Craiova, Romania
Y1 - 2022
PY - 2022
VL - 19
IS - 6
SP - 93
EP - 110
KW - Commutative ring
KW - Ideal
KW - BCK-algebra
KW - residuated lattice
KW - MV-algebra
KW - Boolean algebra
KW - Heyting algebra
KW - Chang property
KW - block codes
DO - 10.22111/ijfs.2022.7213
N2 - 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 $A=\mathbb{Z} _{k_{1}}\times \mathbb{Z}_{k_{2}}\times ...\times \mathbb{Z}_{k_{r}},$ where $k_{i}=p_{i}^{\alpha _{i}}$ and $p_{i}$ a prime number, for all $i\in \{1,2,...,r\}$, 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.
UR - https://ijfs.usb.ac.ir/article_7213.html
L1 - https://ijfs.usb.ac.ir/article_7213_43bce92b52b9a67d396d63a83a1dbcd3.pdf
ER -