Linearly disjoint

In mathematics, algebras "A", "B" over a field "k" inside some field extension formula_1 of "k" are said to be linearly disjoint over "k" if the following equivalent conditions are met:
Note that, since every subalgebra of formula_1 is a domain, (i) implies formula_7 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 formula_7 is a domain then it is a field and "A" and "B" are linearly disjoint. However, there are examples where formula_7 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 formula_1 generated by formula_11, resp. are linearly disjoint over "k". (cf. tensor product of fields)
Suppose "A", "B" are linearly disjoint over "k". If formula_12, formula_13 are subalgebras, then formula_14 and formula_15 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.)