Linearly disjoint

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

• (i) The map ${\displaystyle A\otimes _{k}B\to AB}$ induced by ${\displaystyle (x,y)\mapsto xy}$ is injective.
• (ii) Any k-basis of A remains linearly independent over B.
• (iii) If ${\displaystyle u_{i},v_{j}}$ are k-bases for A, B, then the products ${\displaystyle u_{i}v_{j}}$ are linearly independent over k.

Note that, since every subalgebra of ${\displaystyle \Omega }$ is a domain, (i) implies ${\displaystyle A\otimes _{k}B}$ 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 ${\displaystyle A\otimes _{k}B}$ is a domain then it is a field and A and B are linearly disjoint. However, there are examples where ${\displaystyle A\otimes _{k}B}$ 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 ${\displaystyle \Omega }$ generated by ${\displaystyle A,B}$, resp. are linearly disjoint over k. (cf. tensor product of fields)

Suppose A, B are linearly disjoint over k. If ${\displaystyle A'\subset A}$, ${\displaystyle B'\subset B}$ are subalgebras, then ${\displaystyle A'}$ and ${\displaystyle B'}$ 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