Associated Malcev ring of an alternative ring

For rings
Suppose $$M$$ is an alternative ring with multiplication $$*$$. The associated Malcev ring of $$M$$ is a Malcev ring defined as follows:


 * The underlying set and additive group structure are the same as those of $$M$$.
 * The Malcev multiplication is defined as the defining ingredient::additive commutator for $$M$$, i.e., $$(x,y) \mapsto (x * y) - (y * x)$$.

For algebras
The above definition can be adapted to the case that $$M$$ is an alternative algebra over a commutative unital ring $$R$$. In this case, we construct the Malcev algebra of $$M$$ using the same recipe as above, but we now additionally have a $$R$$-module structure.

Note that the underlying Malcev ring structure remains the same regardless of what commutative unital ring we consider $$M$$ as an algebra over.

Related notions

 * Associated Lie ring of an associative ring