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
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