# Quasivarieties of commutative rings

## DOI:

https://doi.org/10.26577/jmmcs-2018-1-485## Keywords:

quasi-equational theory, undecidable theory, quasi-identity, quasivariety, quasiequational basis, independent quasi-equational bases, recursive independent quasi-equational basis, commutative ring with unit## Abstract

The work is devoted to the problem of undecidability of quasi-equational theories and to the

problem of finite axiomatizability. In 1966, Tarsky poded the following problem: Is there an algorithm

deciding if the equational theory of a finite set of finite structures is finitely axiomatizabile?

In 1986, Maltsev asked the following question: Are there finitely based semigroups, groups, and

rings with the undecidable equational theory? Nurakunov A.M. (Nurakunov, 2012) established

the existence of continuum many quasivarieties of unars, for which the quasi-equational theory is

undecidable and the finite membership problem is also undecidable. In in the paper (Basheyeva,

2017), some results in this direction are obtained for graphs, differential gruppoids, and pointed

Abelian groups. In the present paper, we prove analogous results for the variety of commutative

rings with unit. We prove also that the quasivariety of commutative rings with unit contains continuum

many of subquasivarieties with the undecidable quasi-equational theoryб for which the

finite membership problem is also undecidable. Apart from trhat, we prove here that the quasivariety

of commutative rings with unit contains continuum many subquasivarieties which have an

!-independent quasi-equational basis but does not have an indepen

## References

Symbolic Logic. 57 (1992): 179–192.

[2] Baker, Kirby. "Finite equational bases for finite algebras in congruence-distributive equational classes."Advances in Math.

24 (1977): 207–243.

[3] Basheyeva, Ainur, and Nurakunov Anvar, and Schwidefsky Marina, and Zamojska-Dzienio Anna. "Lattices of subclasses.

III."Siberian Electronic Mathematical Reports. 14 (2017): 252–263.

[4] Birkgoff, Garret. "On the structure of abstract algebras."Proc. Cambridge Philos. Soc. 31 (1935): 433 – 454.

[5] Gorbunov, Viktor. "Covers in lattices of quasivarieties and independent axiomatizability."Algebra and Logic. 16 (1977):

340–369.

[6] Gorbunov, Viktor. Algebraic Theory of Quasivarieties. New York: Plenum, 1998.

[7] Cohen, Daniel. "On the laws of a metabelian variety."J. Alg. 5 (1967): 267–273.

[8] Chin, Luogeng Hua, and Tarski Аlfred. "Distributive and modular lows in the arithmetic of relation algebras."University

of California Publications in Mathematics 9 (1951): 341-384.

[9] J´onsson, Bjarni. "Algebras whose congruence lattices are distributive."Mathematica Scandinavica 21 (1967): 110–121.

[10] J´onsson, Bjarni. "Equational classes of lattices."Mathematica Scandinavica 22 (1968): 187–196.

[11] Kartashov, Vladimir. "Quasivarieties of unary algebras with a finite number of cycles."Algebra and Logic. 19 (1980):

106–120.

[12] Kartashova, Anna. "Antivarieties of unars."Algebra and Logic. 50 (2011): 357–364

[13] Kravchenko, Aleksandr. "Complexity of quasivariety lattices for varieties of unary algebras. II."Siberian Electronic Mathematical

Reports. 13 (2016): 388–394.

[14] Kravchenko, Aleksandr, and Anvar Nurakunov, and Marina Schwidefsky. "Complexity of quasivariety lattices. I. Covers

and independent axiomatizability."manuscript, 2017.

[15] Kravchenko, Aleksandr, and Anvar Nurakunov, and Marina Schwidefsky. "On quasi-equational bases for differential

groupoids and unary algebras."Siberian Electronic Mathematical Reports. 14 (2017): 1330–1337

[16] Kravchenko, Aleksandr, and Andrei Yakovlev. "Quasivarieties of graphs and independent axiomatizability."Siberian Electronic

Mathematical Reports. 20 (2017): 80–89

[17] Kruse, Robert. "Identities satisfied by a finite ring."Journal of Algebra. 26 (1973): 298—318.

[18] Lyndon, Rodger. "Two notes on nilpotent groups."Proc. Amer. LVTcztk. Sot. 3 (1952): 579-583.

[19] L’vov, I.V. "Varieties of associative rings, I, II"Algebra and logika 12 (1973): 267–298, 667–668, 735.

[20] Maltsev, Anatolij. "Universally axiomatizable subclasses of locally finite classes of models."Siberian Math. J. 8 (1967):

764–770.

[21] Medvedev N.Y. "Quasivarieties of l-groups and groups". Siberian Math. J. 26 (1985): 717–723.

[22] McKenzie, Ralf. "Tarski’s finite basis problem is undecidable."International Journal of Algebra and Computation. V. 6

(1996): 49-104.

[23] Murskii, Vadim. "Examples of varieties of semigroups". Algebra and logica. 3 (1968): 423–427.

[24] Neumann, Bernhard. Varieties of Groups. Berlin, Springer-Verlag, Ergeb. d. Math. B. 37, 1967.

[25] Nurakunov, Аnvar. "Unreasonable lattices of quasivarieties."International Journal of Algebra and Computation. V. 22

(2012): 1-17.

[26] Nurakunov, Аnvar. Quasi-indentities of relatively distributive and relatively cocontinuous quasivarieties of algebras. Darmstadt:

Arbeitstangung Allgemeine Algebra, 1995.

[27] Semenova, Marina, Anna Zamojska-Dzienio. "Lattices of subclasses"Siberian Math. J., 53 (2012): 889–905.

[28] Sizyi, Sergei. "Quasivarieties of graphs."Siberian Math. J. 35 (1994): 783–794.

[29] Specht, Wilhelm. "Gesetze in Ringen. I."Math. Z. 52 (1950): 557–589.

[30] Schwidefsky, Marina, Anna Zamojska-Dzienio. "Lattices of subclasses, II"Internat. J. Algebra Comput., 24 (2014): 1099-

1126.

[31] Tarski, Аlfred. "Equational logic and equational theories of algebras."Contrib. Math. Logic. 8 (1966): 275–288.

[32] Tarski, Аlfred. "Some methodological results concerning the calculus of relations."J. Symbolic Logic. 18 (1953): 188–189.

[33] Tarski, Аlfred., Givant S. A Formalization of set theory without variables AMS: Colloquium publications, Providence,

Rhode Island. 41 (1987).

[34] Tropin, Mihail. "Finite pseudo-Boolean and topological algebras not having an independent basis of quasiindentities."

Algebra and Logic. 27 (1988): 79–99.

[35] Tumanov, Vladimir. "Finite lattice with independent quasi-equational basises"Math. notes, 36 (1984): 811–815.

[36] Fedorov, Aleksandr . "Quasi-identities of a free 2-nilpotent group"Math. notes, 40 (1986): 837–841.

[37] Kleiman, Y. (1982) O tojdestvax v gruppax [On indentities in groups]. Tr. Mosk. Mat. obs. - Proseeding of Moskow

mathematical sosiety vol.44, pp. 62–108.

[38] Maltsev, A. (1939) O vkluchenyi assosiativnykh sistem v gruppy [On including of associative systems in groups]. Mat.

sbornik - Math. collection, vol. 6, no. 2, pp.187–189.

[39] Maltsev, A. (1966) O nekotorykh pogranichnikh voprosakh algebry i matematicheskoi logiki [Some boundary questions

of algebra and Mathematical logic]. Trudy kongressa matematikov - Proseeding of mathematic congress (Moskva, 1966),

М.: Мir, pp.217–231.

## Downloads

## How to Cite

*Journal of Mathematics, Mechanics and Computer Science*,

*97*(1), 54–66. https://doi.org/10.26577/jmmcs-2018-1-485