In mathematics, the direct product of a collection of algebraic structures (such as groups, rings, or vector spaces) is a structure of the same type constructed by combining the given structures in a specific way, described below. Its underlying set is the Cartesian product of the underlying sets of the given structures.

The direct sum of a collection of structures agrees with the direct product in some but not all cases. A direct product is an example of a product in a category, whereas a direct sum is an example of a coproduct.

Examples

• is thought of as the set of real numbers

• is thought of as the group

• is thought of as the ring

• is a field does not have a multiplicative inverse.

That relies on the direct product being associative up to isomorphism of the same kind. The direct product is also commutative of the same kind. Even the direct product of infinitely many algebraic structures can be talked about; for example, the direct product of countably

Direct product of groups

In group theory For abelian groups that are written additively, it may also be called the direct sum of two groups

It is defined as follows:

• the set of the elements of the new group is the Cartesian product

The construction gives a new group, which has a normal subgroup).

A relaxation of those conditions by requiring only one subgroup to be normal gives the semidirect product.

With a direct product, some natural group homomorphisms are called the coordinate functions.

-tuples that is the trivial group

Direct product of modules

The direct product for modules (not to be confused with the tensor product) is very similar to the one that is defined for groups above by using the Cartesian product with the operation of addition being componentwise, and the scalar multiplication, Euclidean space

is canonically isomorphic to the direct sum The direct sum and the direct product are not isomorphic for infinite indices for which the elements of a direct sum are zero for all but for a finite number of entries. They are dual in the sense of category theory: the direct sum is the coproduct, and the direct product is the product.

).

Topological space direct product

The direct product for a collection of topological spaces some index set

Defining the topology is a little tricky. For finitely many factors, it is the obvious and natural thing to do: simply take as a basis

That topology is called the product topology (disjoint unions of open intervals), the basis for that topology would consist of all disjoint unions of open rectangles in the plane (as it turns out, it coincides with the usual metric topology).

The more natural-sounding topology would be, in this case, to take products of infinitely many open subsets as before, which yields a somewhat interesting topology, the box topology. However, it is not too difficult to find an example of bunch of continuous component functions whose product function is not continuous (see the separate entry box topology for an example and more). The problem that makes the twist necessary is ultimately rooted in the fact that the intersection of open sets is guaranteed to be open only for finitely many sets in the definition of topology.

Products (with the product topology) are nice with respect to preserving properties of their factors; for example, the product of Hausdorff spaces is Hausdorff, the product of connected spaces is connected, and the product of compact spaces is compact. That last one, called Tychonoff's theorem, is yet another equivalence to the axiom of choice.

For more properties and equivalent formulations, see product topology.

Direct product of binary relations

On the Cartesian product of two sets with binary relations are both reflexive, irreflexive, transitive, symmetric, or antisymmetric will be also. Similarly, totality If the properties are combined, that also applies for being a preorder and being an equivalence relation are connected relations

Direct product in universal algebra

is a fixed signature is an indexed family algebras, the direct product algebra defined as follows:

It is a surjective homomorphism

contains only one binary operation the above definition of the direct product of groups Similarly, the definition of the direct product of modules is subsumed here.

Categorical product

The direct product can be abstracted to an arbitrary category, a product together with morphisms. .

In the special case of the category of groupsth coordinate.

Internal and external direct product

Some authors draw a distinction between an internal direct product and an external direct product is the internal can be referred to as the external.

See also

• Tensor product and Kronecker product also use the symbol ⊗ to produce a larger space

Notes

References

ru:Прямое произведение#Прямое произведение групп