# Product ring

Algebraic structure → Ring theoryRing theory |
---|

In mathematics, it is possible to combine several rings into one large **product ring**. This is done by giving the Cartesian product of a (possibly infinite) family of rings coordinatewise addition and multiplication. The resulting ring is called a **direct product** of the original 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),

- [math]\displaystyle{ n=p_1^{n_1} p_2^{n_2}\cdots\ p_k^{n_k}, }[/math]

where the *p _{i}* are distinct primes, then

**Z**/

*n*

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

- [math]\displaystyle{ \mathbf{Z}/p_1^{n_1}\mathbf{Z} \ \times \ \mathbf{Z}/p_2^{n_2}\mathbf{Z} \ \times \ \cdots \ \times \ \mathbf{Z}/p_k^{n_k}\mathbf{Z}. }[/math]

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.

When *I* is finite, the underlying additive group of Π_{i∈I} *R*_{i} coincides with the direct sum of the additive groups of the *R*_{i}. In this case, some authors call *R* the "direct sum of the rings *R*_{i}" and write ⊕_{i∈I} *R*_{i}, but this is incorrect from the point of view of category theory, since it is usually not a coproduct in the category of rings: for example, when two or more of the *R*_{i} are nonzero, the inclusion map *R _{i}* →

*R*fails to map 1 to 1 and hence is not a ring homomorphism.

(A finite coproduct in the category of commutative (associative) algebras over a commutative ring is a tensor product of algebras. A coproduct in the category of algebras is a free product of algebras.)

Direct products are commutative and associative (up to isomorphism), meaning that it doesn't matter in which order one forms the direct product.

If *A _{i}* is an ideal of

*R*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 two or more non-zero rings always has nonzero zero divisors: if *x* is an element of the product whose coordinates are all zero except *p _{i}*(

*x*), and

*y*is an element of the product with all coordinates zero except

*p*(

_{j}*y*) where

*i*≠

*j*, then

*xy*= 0 in the product ring.

## See also

## Notes

## References

- Herstein, I.N. (2005),
*Noncommutative rings*(5th ed.),*Cambridge University Press*, ISBN 978-0-88385-039-8, https://archive.org/details/noncommutativeri0000hers - Lang, Serge (2002),
*Algebra*, Graduate Texts in Mathematics,**211**(Revised third ed.), New York: Springer-Verlag, p. 91, ISBN 978-0-387-95385-4