# Exterior square of a Lie ring

From Groupprops

## Definition

Suppose is a Lie ring. The **exterior square** of , denoted or , can be defined in the following equivalent ways:

- It is the exterior product of with itself, viewing it as two copies of itself in itself (in general, the exterior product is defined for two possibly equal ideals inside a Lie ring).
- It is the derived subring of any Schur covering Lie ring of . Note that the Schur covering Lie rings need not be isomorphic Lie rings, but they are isoclinic Lie rings, so the definition is independent of the choice of Schur covering Lie ring.
- If where is a free Lie ring and is an ideal in , it is, up to isomorphism, the same as .