Editing Direct sum of modules (section)

===Direct sum of algebras===

A direct sum of [[Algebra over a field|algebras]] <math>X</math> and <math>Y</math> is the direct sum as vector spaces, with product 
:<math>(x_1 + y_1) (x_2 + y_2) = (x_1 x_2 + y_1 y_2).</math>
Consider these classical examples:
:<math>\mathbf{R} \oplus \mathbf{R}</math> is [[Ring isomorphism|ring isomorphic]] to [[split-complex number]]s, also used in [[interval analysis]].
:<math>\mathbf{C} \oplus \mathbf{C}</math> is the algebra of [[tessarine]]s introduced by [[James Cockle (lawyer)|James Cockle]] in 1848.
:<math>\mathbf{H} \oplus \mathbf{H},</math> called the [[split-biquaternion]]s, was introduced by [[William Kingdon Clifford]] in 1873.
[[Joseph Wedderburn]] exploited the concept of a direct sum of algebras in his classification of [[hypercomplex number]]s. See his ''Lectures on Matrices'' (1934), page 151.
Wedderburn makes clear the distinction between a direct sum and a direct product of algebras: For the direct sum the field of scalars acts jointly on both parts: <math>\lambda (x \oplus y) = \lambda x \oplus \lambda y</math> while for the direct product a scalar factor may be collected alternately with the parts, but not both:
<math>\lambda (x,y) = (\lambda x, y) = (x, \lambda y).</math>
[[Ian R. Porteous]] uses the three direct sums above, denoting them <math>^2 R,\ ^2 C,\ ^2 H,</math> as rings of scalars in his analysis of ''Clifford Algebras and the Classical Groups'' (1995).

The construction described above, as well as Wedderburn's use of the terms {{em|direct sum}} and {{em|direct product}} follow a different convention than the one in [[category theory]]. In categorical terms, Wedderburn's {{em|direct sum}} is a [[Product (category theory)|categorical product]], whilst Wedderburn's {{em|direct product}} is a [[Coproduct|coproduct (or categorical sum)]], which (for commutative algebras) actually corresponds to the [[tensor product of algebras]].