Linearly disjoint

From formulasearchengine
Jump to navigation Jump to search

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:

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


Template:Algebra-stub