Geodesic
A correspondence between commutative rings and Jordan loops
Algebra i logika, Tome 58 (2019) no. 6, pp. 741-768.

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We show that there is a one-to-one correspondence (up to isomorphism) between commutative rings with unity and metabelian commutative loops belonging to a particular finitely axiomatizable class. Based on this correspondence, it is proved that the sets of identically valid formulas and of finitely refutable formulas of a class of finite nonassociative commutative loops (and of many of its other subclasses) are recursively inseparable. It is also stated that nonassociative commutative free automorphic loops of any nilpotency class have an undecidable elementary theory.
Keywords: commutative ring with unity, metabelian commutative loop, finitely axiomatizable class, undecidability of elementary theory, recursively inseparable sets. 
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V. I. Ursu. A correspondence between commutative rings and Jordan loops. Algebra i logika, Tome 58 (2019) no. 6, pp. 741-768. http://geodesic.mathdoc.fr/item/AL_2019_58_6_a4/

[1] Yu. L. Ershov, I. A. Lavrov, A. D. Taimanov, M. A. Taitslin, “Elementarnye teorii”, Uspekhi matem. n., 20:4(124) (1965), 37–108 | MR | Zbl

[2] A. Tarski, A. Mostowski, R. M. Robinson, Undecidable theories, Stud. Logic Found. Math., North-Holland Publ. Co., Amsterdam, 1953 | MR | Zbl

[3] A. I. Maltsev, “Ob odnom sootvetstvii mezhdu koltsami i gruppami”, Matem. sb., 50(92):3 (1960), 257–266 | Zbl

[4] A. I. Maltsev, “Nerazreshimost elementarnoi teorii konechnykh grupp”, Dokl. AN SSSR, 138:4 (1961), 771–774 | Zbl

[5] A. I. Maltsev, “Effektivnaya neotdelimost mnozhestva tozhdestvenno istinnykh i mnozhestva konechno oproverzhimykh formul nekotorykh elementarnykh teorii”, Dokl. AN SSSR, 139:4 (1961), 802–805 | Zbl

[6] V. I. Ursu, “Rekursivnaya neotdelimost mnozhestva tozhdestvenno istinnykh i mnozhestva konechno oproverzhimykh formul nekotorykh elementarnykh teorii mnogoobrazii”, Sib. matem. zh., 41:3 (2000), 696–711 | MR | Zbl

[7] V. D. Belousov, Osnovy teorii kvazigrupp i lup, Nauka, M., 1967

[8] R. H. Bruck, A survey of binary systems, Ergeb. Math. Grenzgeb. Neue Folge, 20, Springer-Verlag, Berlin–Göttingen–Heidelberg, 1958 | MR | Zbl

[9] A. V. Kovalski, V. I. Ursu, “Ekvatsionalnaya teoriya nilpotentnoi $A$-lupy”, Algebra i logika, 49:4 (2010), 479–497 | MR

[10] R. H. Bruck, L. J. Paige, “Loops whose inner mappings are automorphisms”, Ann. Math. (2), 63 (1956), 308–323 | DOI | MR | Zbl