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 a family of algebraic objects is
given by the Cartesian product of their underlying
sets, with algebraic operations
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. Neither the category
product nor the coproduct requires commutativity.
For example, the coproduct of groups
is their free product. 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 index sets. 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 vector space tensor product, in which case the dimensions multiply instead of add. Here, "direct" may be used to distinguish it from the external tensor product.
