In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the product in category theory, which formalizes these notions.

Examples are the product of sets (see Cartesian product), groups (described below), the product of rings and of other algebraic structures. The product of topological spaces is another instance.

There is also the direct sum – in some areas this is used interchangeably, in others it is a different concept.

Examples

If we think of as the set of real numbers, then the direct product is precisely just the cartesian product, .

If we think of as the group of real numbers under addition, then the direct product still consists of . The difference between this and the preceding example is that is now a group. We have to also say how to add their elements. This is done by letting .

In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. Numbers such as the real or complex numbers can be multiplied according to elementary arithmetic. On the other hand, matrices are arrays of numbers, so there is no unique way to define "the" multiplication of matrices. As such, in general the term "matrix multiplication" refers to a number of different ways to multiply matrices. The key features of any matrix multiplication include: the number of rows and columns the original matrices have (called the "size", "order" or "dimension"), and specifying how the entries of the matrices generate the new matrix.

Like vectors, matrices of any size can be multiplied by scalars, which amounts to multiplying every entry of the matrix by the same number. Similar to the entrywise definition of adding or subtracting matrices, multiplication of two matrices of the same size can be defined by multiplying the corresponding entries, and this is known as the Hadamard product. Another definition is the Kronecker product of two matrices, to obtain a block matrix.

In mathematics, it is possible to combine several rings into one large product ring. This is done as follows: if I is some index set and R_i is a ring for every i in I, then the cartesian product Π_{i ∈ I}R_i can be turned into a ring by defining the operations coordinate-wise.

The resulting ring is called a direct product of the rings R_i. The direct product of finitely many rings coincides with the direct sum of rings.

Examples

An important example is the ring Z/nZ of integers modulo n. If n is written as a product of prime powers (see fundamental theorem of arithmetic):

where the p_i are distinct primes, then Z/nZ is naturally isomorphic to the product ring

This follows from the Chinese remainder theorem.

Properties

If R = Π_{i ∈ I}R_i is a product of rings, then for every i in I we have a surjective ring homomorphism p_i: R → R_i which projects the product on the ith coordinate. The product R, together with the projections p_i, has the following universal property:

This shows that the product of rings is an instance of products in the sense of category theory. However, despite also being called the direct sum of rings when I is finite, the product of rings is not a coproduct in the sense of category theory. In particular, if I has more than one element, the inclusion map R_i → R is not ring homomorphism as it does not map the identity in R_i to the identity in R.