The product of finitely many rings R_{1},...,R_{k} is also written as R_{1} × R_{2} × ... × R_{k}.
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):
where the p_{i} are distinct primes, then Z/nZ is naturally isomorphic to the product ring
If R = Π_{i in 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.
If A is a (left, right, twosided) ideal in R, then there exist (left, right, twosided) ideals A_{i} in R_{i} such that A = Π_{i in I} A_{i}. 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 A_{i} are equal to R_{i} and the remaining A_{i} is a prime ideal in R_{i}.
An element x in R is a unit if and only if all of its components are units, i.e. if and only if p_{i}(x) is a unit in R_{i} for every i in I. The group of units of R is the product of the groups of units of R_{i}.
A product of more than one nonzero rings always has zero divisors: if x is an element of the product all of whose coordinates are zero except p_{i}(x), and y is an element of the product with all coordinates zero except p_{j}(y) (with i ≠ j), then xy = 0 in the product ring.
Search Encyclopedia

Featured Article
