# Linearly disjoint

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In mathematics, algebras *A*, *B* over a field *k* inside some field extension of *k* (e.g., universal field) 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).

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

## References

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