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2.39 (1966)

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Does there exist a non-finitely-axiomatizable variety of
$\qquad$ a) (H. Neumann) groups?
$\qquad$ b) of associative rings (the Specht problem)?
$\qquad$ c) Lie rings?

Contributor: A. I. Mal’cev

Progress

a) Yes, there does (S. I. Adian, Math. USSR–Izv., 4 (1970), 721–739; A. Yu. Ol'shanskiĭ, Math. USSR–Izv., 4 (1970), 381–389).

b) Yes, there does (A. Ya. Belov, Fundam. Prikl. Mat., 5, no. 1 (1999), 47–66 (Russian), Sb. Math., 191, no. 3–4 (2000), 329–340; A. V. Grishin, Fundam. Prikl. Mat., 5, no. 1 (1999), 101–118 (Russian); V. V. Shchigolev, Fundam. Prikl. Mat., 5, no. 1 (1999), 307–312 (Russian)). But every variety of associative algebras over a field of characteristic 0 has a finite basis of identities (A. R. Kemer, Algebra and Logic, 26, no. 5 (1987), 362–397).

c) Yes, there does (M. R. Vaughan-Lee, Quart. J. Math., 21 (1970), 297–308).

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