Connections between commutative rings and some algebras of logic

Document Type : Research Paper


1 Faculty of Mathematics and Computer Science, Ovidius University, Bd. Mamaia 124, 900527, Constanta, Romania

2 Faculty of Science, University of Craiova, A.I. Cuza Street, 13, 200585, Craiova, Romania


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.


[1] H. A. S. Abujabal, M. Aslam, A. B. Thaheem, A representation of bounded commutative BCK-algebras, International Journal of Mathematics and Mathematical Sciences, 19(4) (1996), 733-736.
[2] R. Balbes, P. Dwinger, Distributive lattices, Columbia, Missouri: University of Missouri Press. XIII, 1974.
[3] L. P. Belluce, A. Di Nola, Commutative rings whose ideals form an MV-algebra, Mathematical Logic Quarterly, 55(5) (2009), 468-486.
[4] L. P. Belluce, A. Di Nola, E. Marchioni, Rings and Gödel algebras, Algebra University, 64(1-2) (2010), 103-116.
[5] R. L. Blair, Ideal lattices and the structure of rings, Transactions of the AMS - American Mathematical Society, 75 (1953), 136-153.
[6] K. Blount, C. Tsinakis, The structure of residuated lattices, International Journal of Algebra and Computation, 13(4) (2003), 437-461.
[7] D. Busneag, D. Piciu, Lectii de algebra, Ed. Universitaria, Craiova, 2002.
[8] I. Chajda, H. Länger, Commutative rings whose ideal lattices are complemented, Asian-European Journal of Mathematics, 3 (2019), DOI:10.1142/S1793557119500396.
[9] C. C. Chang, Algebraic analysis of many-valued logic, Transactions of the AMS - American Mathematical Society, 88 (1958), 467-490.
[10] R. Cignoli, I. M. L. D’Ottaviano, D. Mundici, Algebraic foundations of many-valued reasoning, Trends in LogicStudia Logica Library 7, Dordrecht: Kluwer Acad. Publ, 2000, DOI:10.1007/978-94-015-9480-6.
[11] R. P. Dilworth, Abstract residuation over lattices, Bulletin of the AMS - American Mathematical Society, 44 (1938), 262-268.
[12] C. Flaut, S. Hoskova-Mayerova, A. Borumand Saeid, R. Vasile, Wajsberg algebras of order n(n ≤ 9), Neural Computing and Applications, 32 (2020), 13301-13312.
[13] C. Flaut, R. Vasile, Wajsberg algebras arising from binary block codes, Soft Computing, 24 (2020), 6047-6058.
[14] J. M. Font, A. J. Rodriguez, A. Torrens, Wajsberg algebras, Stochastica, 8(1) (1984), 5-30.
[15] B. Van Gasse, G. Deschrijver, C. Cornelis, E. Kerre, Filters of residuated lattices and triangle algebras, Information Sciences, 180(16) (2010), 3006-3020.
[16] I. N. Hernstein, Topics in algebra, 2end edition, John Wiley and Son, New York, 1975. [17] U. Höhle, S. E. Rodabaugh, Mathematics of fuzzy sets: Logic, topology and measure theory, Springer, Berlin, 1999.
[18] A. Iorgulescu, Algebras of logic as BCK algebras, A.S.E., Bucharest, 2009.
[19] J. H. Lint, Introduction to coding theory, third edition, Graduate Texts in Mathematics, 86, Springer Verlag, Berlin, 1999.
[20] J. Meng, Y. B. Jun, BCK-algebras, Kyung Moon Sa Co. Seoul, Korea, 1994.
[21] D. Mundici, MV-algebras-a short tutorial, Department of Mathematics Ulisse Dini, University of Florence, 2007.
[22] D. Piciu, Algebras of fuzzy logic, Editura Universitaria, Craiova, 2007.
[23] S. V. Tchoffo Foka, M. Tonga, Rings and residuated lattices whose fuzzy ideals form a Boolean algebra, Soft Computing, 26 (2022), 535-539.
[24] E. Turunen, Mathematics behind fuzzy logic, Physica-Verlag Heidelberg, 1999.
[25] M. Ward, R. P. Dilworth, Residuated lattices, Transactions of the AMS - American Mathematical Society, 45 (1939), 335-354.