# Product of rings

In abstract algebra, 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 in I}

*R*

_{i}can be turned into a ring by defining the operations coordinatewise, i.e.

- (
*a*) + (_{i}*b*) = (_{i}*a*+_{i}*b*)_{i} - (
*a*) · (_{i}*b*) = (_{i}*a*·_{i}*b*)_{i}

The product of finitely many rings *R*_{1},...,*R*_{k} is also written as *R*_{1} × *R*_{2} × ... × *R*_{k}.

### Examples

The most important example is the ring **Z**/*n***Z** 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**/

*n*

**Z**is naturally isomorphic to the product ring

This follows from the Chinese remainder theorem.

### Properties

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*which projects the product on the

_{i}*i*-th coordinate. The product

*R*, together with the projections

*p*, has the following universal property:

_{i}- if
*S*is any ring and*f*:_{i}*S*`->`*R*is a ring homomorphism for every_{i}*i*in*I*, then there exists*precisely one*ring homomorphism*f*:*S*`->`*R*such that*f*o*p*=_{i}*f*for every_{i}*i*in*I*.

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 *A _{i}* in

*R*such that

_{i}*A*= Π

_{i in I}

*A*. Conversely, every such product of ideals is an ideal in

_{i}*R*.

*A*is a prime ideal in

*R*if and only if all but one of the

*A*are equal to

_{i}*R*and the remaining

_{i}*A*is a prime ideal in

_{i}*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*for every

_{i}*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 non-zero 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.