The most 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):
Examples
where the pi are distinct primes,
then Z/nZ is naturally isomorphic to the product ring
If R = Πi in I Ri is a product of rings, then for every i in I we have a surjective ring homomorphism pi : R -> Ri which projects the product on the i-th coordinate. The product R, together with the projections pi, has the following universal property:
Properties
This shows that the product of rings is an instance of products in the sense of category theory.
If A is a (left, right, two-sided) ideal in R, then there exist (left, right, two-sided) ideals Ai in Ri such that A = Πi in I Ai. Conversely, every such product of ideals is an ideal in R. A is a prime ideal in R if and only if all but one of the Ai are equal to Ri and the remaining Ai is a prime ideal in Ri.
An element x in R is a unit if and only if all of its components are units, i.e. if and only if pi(x) is a unit in Ri for every i in I. The group of units of R is the product of the groups of units of Ri.
A product of more than one non-zero rings always has zero divisors: if x is an element of the product all of whose coordinates are zero except pi(x), and y is an element of the product with all coordinates zero except pj(y) (with i ≠ j), then xy = 0 in the product ring.