mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

algebras In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...

''A'', ''B'' over a field ''k'' inside some

field extension In mathematics, particularly in algebra, a field extension is a pair of fields K \subseteq L, such that the operations of ''K'' are those of ''L'' restricted to ''K''. In this case, ''L'' is an extension field of ''K'' and ''K'' is a subfield of ...

\Omega

of ''k'' are said to be linearly disjoint over ''k'' if the following equivalent conditions are met: *(i) The map

A \otimes_k B \to AB

induced by

(x, y) \mapsto xy

injective In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...

. *(ii) Any ''k''- basis of ''A'' remains

linearly independent In the theory of vector spaces, a set of vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be . These concep ...

over ''B''. *(iii) If

u_i, v_j

are ''k''-bases for ''A'', ''B'', then the products

u_i v_j

are linearly independent over ''k''. Note that, since every subalgebra of

\Omega

is a domain, (i) implies

A \otimes_k B

is a domain (in particular reduced). Conversely if ''A'' and ''B'' are fields and either ''A'' or ''B'' is an

algebraic extension In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, every element of is a root of a non-zero polynomial with coefficients in . A field extens ...

of ''k'' and

A \otimes_k B

is a domain then it is a field and ''A'' and ''B'' are linearly disjoint. However, there are examples where

A \otimes_k B

is a domain but ''A'' and ''B'' are not linearly disjoint: for example, ''A'' = ''B'' = ''k''(''t''), the

field of rational functions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the fiel ...

over ''k''. One also has: ''A'', ''B'' are linearly disjoint over ''k''

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

the subfields of

\Omega

generated by

A, B

, resp. are linearly disjoint over ''k''. (cf.

Tensor product of fields In mathematics, the tensor product of two field (mathematics), fields is their tensor product of algebras, tensor product as algebra over a field, algebras over a common subfield (mathematics), subfield. If no subfield is explicitly specified, t ...

) Suppose ''A'', ''B'' are linearly disjoint over ''k''. If

A' \subset A

B' \subset B

are subalgebras, then

A'

and

B'

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.)

References

* P.M. Cohn (2003). Basic algebra Algebra {{algebra-stub

See also

References