The direct product is defined for a number of classes of algebraic objects, including sets, groups, rings, and modules. In each case, the direct product of an algebraic object is given by the Cartesian product of its elements, considered as sets, and its algebraic operations are defined componentwise. For instance, the direct product of two vector spaces of dimensions and is a vector space of dimension .

Direct products satisfy the property that, given maps and , there exists a unique map given by . The notion of map is determined by the category, and this definition extends to other categories such as topological spaces. Note that no notion of commutativity is necessary, in contrast to the case for the coproduct. In fact, when and are Abelian, as in the cases of modules (e.g., vector spaces) or Abelian groups (which are modules over the integers), then the direct sum is well-defined and is the same as the direct product. Although the terminology is slightly confusing because of the distinction between the elementary operations of addition and multiplication, the term "direct sum" is used in these cases instead of "direct product" because of the implicit connotation that addition is always commutative.

Note that direct products and direct sums differ for infinite indices. An element of the direct sum is zero for all but a finite number of entries, while an element of the direct product can have all nonzero entries.

Some other unrelated objects are sometimes also called a direct product. For example, the tensor direct product is the same as the tensor product, in which case the dimensions multiply instead of add. Here, "direct" may be used to distinguish it from the external tensor product.

This entry contributed by Todd Rowland

Korn, G. A. and Korn, T. M. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill, p. 393, 1968.

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Rowland, Todd. "Direct Product." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/DirectProduct.html