The **direct product** takes the **Cartesian product** \({A\times B}\) of sets, i.e. the ordered pairs of elements \({(a,b)}\), and applies all operations component-wise; e.g. for a group we define \({(a,b)+(c,d)\equiv(a+c,b+d)}\). Note that this approach cannot be taken in all categories; for example, a new field cannot be obtained from the direct product of two fields, since \({(0,a)}\) is distinct from \({\mathbf{0}}\) and has no multiplicative inverse. The **direct sum** is identical to the direct product except in the case of an infinite number of factors, when the direct sum \({\bigoplus A_{\mu}}\) consists of elements that have only finitely many non-identity terms, while the direct product \({\prod A_{\mu}}\) has no such restriction.

Δ For a finite number of objects, the direct product and direct sum are identical constructions, and these terms are often used interchangeably, along with their symbols \({\times}\) and \({\oplus}\). In particular, since the group operation is usually written like multiplication, we usually write \({G\times H}\); with vector spaces and algebras, where the abelian group operation is written like addition, we instead write \({V\oplus W}\). |

Δ The categorical coproduct for abelian groups and vector spaces is the direct sum, which is then also applied to objects in other categories, potentially causing confusion with the “categorical direct sum” (biproduct) in those other categories, which may be a distinct construction or not exist. |

An additional distinction that can be made is between an **external** sum or product, where the construction of a new object is from given constituent objects, and an **internal** sum or product, which is formed from a given object by recognizing constituent sub-objects within it and noting that the sum or product of the sub-objects is isomorphic to the original object.

Δ In addition to denoting a direct sum, the symbol \({\oplus}\) is sometimes used to denote either the internal direct product of groups or the free product. In general, the above symbols are not always used consistently and it is important to understand exactly what construction is meant in a given situation. |

As we will see, in addition to this “internal/external” distinction, there is a rough logic as well to the distinction between the designations “inner/outer” and “interior/exterior” with regard to products.