Linear disjointness

In mathematics, algebras A, B over a field k inside some field extension of k are said to be linearly disjoint over k if the following equivalent conditions are met:

The map induced by is injective.
Any k-basis of A remains linearly independent over B.
There exists a k-basis of A which remains linearly independent over B.
If are k-bases for A, B, then the products are linearly independent over k.

Note that, since every subalgebra of is a domain, implies is a domain. Conversely if A and B are fields and either A or B is an algebraic extension of k and is a domain then it is a field and A and B are linearly disjoint. However, there are examples where is a domain but A and B are not linearly disjoint: for example, A = B = k, the field of rational functions over k.
One also has: A, B are linearly disjoint over k if and only if the subfields of generated by, resp. are linearly disjoint over k.
Suppose A, B are linearly disjoint over k. If, are subalgebras, then and are linearly disjoint over k. Conversely, if any finitely generated subalgebras of algebras A, B are linearly disjoint, then A, B are linearly disjoint