On a class of Lie rings of 2 x 2 matrices over associative commutative rings

Let k be an arbitrary associative and commutative ring. If A(11), A(12), A(21) are subgroups of the additive group of k such that 2A(11)A(12) subset of A(12), 2A(11)A(21) subset of A(21), A(12)A(21) subset of A(11), then matrices (a(11) a(12) a(21) -a(11)) with a(ij) is an element of A(ij) form a subring of the Lie ring of all 2 x 2 matrices over k whose trace is 0. The paper studies certain properties of these Lie rings. As an application of the notion introduced, the description of Lie rings lying between sl(2)(Z) and sl(2)(K), K an integral quadratic extension of Z, is given.