In mathematics, a product of rings or direct product of rings is a ring that is formed by the Cartesian product of the underlying sets of several rings (possibly an infinity), equipped with componentwise operations. It is a direct product in the category of rings. Since direct products are defined up to an isomorphism, one says colloquially that a ring is the product of some rings if it is isomorphic to the direct product of these rings. For example, the Chinese remainder theorem may be stated as: if and are coprime integers, the quotient ring is the product of and Examples An important example is Z/nZ, the ring of integers modulo n. If n is written as a product of prime powers (see Fundamental theorem of arithmetic), where the pi are distinct primes, then Z/nZ is naturally isomorphic to the product This follows from the Chinese remainder theorem. Properties If is a product of rings, then for every i in I we have a surjective ring homomorphism which projects the product on the i&hairsp;th coordinate. The product R together with the projections pi has the following universal property: if S is any ring and is a ring homomorphism for every i in I, then there exists precisely one ring homomorphism such that for every i in I. This shows that the product of rings is an instance of products in the sense of category theory. When I is finite, the underlying additive group of coincides with the direct sum of the additive groups of the Ri. In this case, some authors call R the "direct sum of the rings Ri" and write , but this is incorrect from the point of view of category theory, since it is usually not a coproduct in the category of rings (with identity): for example, when two or more of the Ri are non-trivial, the inclusion map fails to map 1 to 1 and hence is not a ring homomorphism. (A finite coproduct in the category of commutative algebras over a commutative ring is a tensor product of algebras. A coproduct in the category of algebras is a free product of algebras.) Direct products are commutative and associative up to natural isomorphism, meaning that it doesn't matter in which order one forms the direct product. If Ai is an ideal of Ri for each i in I, then is an ideal of R. If