# Product of rings

In mathematics, 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 ∈ I}

*R*

_{i}can be turned into a ring by defining the operations coordinate-wise.

The resulting ring is called a **direct product** of the rings *R*_{i}. The direct product of finitely many rings coincides with the direct sum of rings.

## Examples

An 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 ∈ 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*p*∘_{i}*f*=*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. However, despite also being called the direct sum of rings when *I* is finite, the product of rings is not a coproduct in the sense of category theory. In particular, if *I* has more than one element, the inclusion map *R _{i}* →

*R*is not ring homomorphism as it does not map the identity in

*R*to the identity in

_{i}*R*.

If *A _{i}* in

*R*is an ideal for each

_{i}*i*in

*I*, then

*A*= Π

_{i ∈ I}

*A*is an ideal of

_{i}*R*. If

*I*is finite, then the converse is true, i.e. every ideal of

*R*is of this form. However, if

*I*is infinite and the rings

*R*are non-zero, 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

_{i}*R*. The ideal

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

*R*if all but one of the

*A*are equal to

_{i}*R*and the remaining

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

_{i}*R*. However, the converse is not true when

_{i}*I*is infinite. For example, the direct sum of the

*R*form an ideal not contained in any such

_{i}*A*, but the axiom of choice gives that it is contained in some maximal ideal which is a fortiori prime.

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.

## See also

## Notes

## References

- Herstein, I.N. (2005) [1968],
*Noncommutative rings*(5th ed.), Cambridge University Press, ISBN 978-0-88385-039-8 - Lang, Serge (2002),
*Algebra*, Graduate Texts in Mathematics,**211**(Revised third ed.), New York: Springer-Verlag, p. 91, ISBN 978-0-387-95385-4, Zbl 0984.00001, MR 1878556