Algebra

Linearly disjoint

In mathematics, algebras A, B over a field k inside some field extension of k 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). Conversely if A and B are fields and either A or B is an algebraic extension of k and is a domain then it is a field and A and B are linearly disjoint. However, there are examples where 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 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.) (Wikipedia).

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Algebraic extension | Field extension | Basis (linear algebra) | Integral domain | If and only if | Tensor product of fields | Mathematics | Field (mathematics) | Algebra over a field | Reduced ring