Linearly disjoint
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:
(i) The map  induced by  is injective. (ii) Any k-basis of A remains linearly independent over B. (iii) If  are k-bases for A, B, then the products  are linearly independent over k.

Note that, since every subalgebra of  is a domain, (i) implies  is a domain (in particular reduced). 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(t), the field of rational functions over k.

One also has: A, B are linearly disjoint over k if and only if subfields of  generated by , resp. are linearly disjoint over k. (cf. Tensor product of fields)

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 (since the condition involves only finite sets of elements.) See also 
Tensor product of fields

References 
 P.M. Cohn (2003). Basic algebra

Algebra