The tensor product of algebras is a special case and generalisation of the tensor product of linear spaces that can be defined directly. We have chosen not to develop the theory of tensor products in general, as we have no need of the more general concept.

Tensor products of real algebras

Certain algebras over a commutative field K admit a decomposition somewhat analogous to the direct sum decompositions of a linear space, but involving the multiplicative structure rather than the additive structure.

Suppose that B and C are subalgebras of a finite-dimensional algebra A over K, the algebra being associative and with unit element, such that

(i) for any b ∈ B, c ∈ C, c b = b c,

(ii) A is generated as an algebra by B and C,

(iii) dim A = dim B dim C.

Then we say that A is the tensor product B ⊕KC over K, the abbreviation B ⊗ C being used when the field K is not in doubt.

Proposition 11.1Let B and C be subalgebras of a finite-dimensional algebra A over K, such that A = B × C, the algebra A being associative and with unit element. Then B ∩ C = K (the field K being identified with the set of scalar multiples of the unit element 1(A))

It is tempting to suppose that the condition B ∩ C = K can be used as an alternative to condition (iii) in the definition.