Variety of Lie rings

As a plain variety
The variety of Lie rings is the variety of algebras with the operator domain consisting of:


 * A binary operation $$+$$
 * A unary operation $$-$$ (prefix symbol)
 * A constant $$0$$
 * A binary operation $$[ \, \ ]$$

such that the following universal identities are satisfied:


 * 1) $$(x + y) + z = x + (y + z)$$
 * 2) $$x + 0 = 0 + x$$
 * 3) $$x + (-x) = (-x) + x = 0$$
 * 4) $$x + y = y + x$$
 * 5) $$[x,y+z] = [x,y] + [x,z]$$
 * 6) $$[x + y,z] = [x,z] + [y,z]$$
 * 7) $$[x,x] = 0$$
 * 8) $$[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0$$

The identities (1)-(4) say that we get an Abelian group under $$+$$, the identities (5) and (6) say the Lie bracket is bilinear, the identity (7) says it is alternating, and the identity (8) is the Jacobi identity.

Properties
A complete listing of the universal algebra-theoretic properties of Lie rings is available at:

Category:Property satisfactions for the variety of Lie rings

Good congruence-related properties

 * Variety of Lie rings is ideal-determined
 * Ideals are subalgebras in the variety of Lie rings