mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

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 ...

''A'', ''B'' over a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

''k'' inside some

field extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' 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; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...

. *(ii) Any ''k''-

basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...

of ''A'' remains

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

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 Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...

, (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, if every element of is a root of a non-zero polynomial with coefficients in . A field ext ...

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 field ...

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" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

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 fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must have the same characteristic and the common subfield is their prime subfie ...

) 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