Gröbner–Shirshov bases for Lie algebras over a commutative algebra

In this paper we establish a Gröbner–Shirshov bases theory for Lie algebras over commutative rings. As applications we give some new examples of special Lie algebras (those embeddable in associative algebras over the same ring) and non-special Lie algebras (following a suggestion of P.M. Cohn (1963) [28]). In particular, Cohnʼs Lie algebras over the characteristic p are non-special when p=2,3,5. We present an algorithm that one can check for any p, whether Cohnʼs Lie algebras are non-special. Also we prove that any finitely or countably generated Lie algebra is embeddable in a two-generated Lie algebra.