Linearly disjoint

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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.)

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References[edit]

  • P.M. Cohn (2003). Basic algebra