Ring homomorphisms Algebraic structures Related structures Algebraic number theory Noncommutative algebraic geometry Free algebra Clifford algebra 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.
For example, the Chinese remainder theorem may be stated as: if m and n are coprime integers, the quotient ring
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.
Direct products are commutative and associative up to natural isomorphism, meaning that it doesn't matter in which order one forms the direct product.
However, if I is infinite and the rings Ri are non-trivial, then the converse is false: the set of elements with all but finitely many nonzero coordinates forms an ideal which is not a direct product of ideals of the Ri.
For example, the direct sum of the Ri form an ideal not contained in any such A, but the axiom of choice gives that it is contained in some maximal ideal which is a fortiori prime.