Product ring

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/nZ 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/nZ 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_i which projects the product on the ith coordinate. The product R, together with the projections p_i, has the following universal property:

if S is any ring and f_i: S → R_i is a ring homomorphism for every i in I, then there exists precisely one ring homomorphism f: S → R such that p_i ∘ f = f_i for every 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_i for each i in I, then A = Π_i∈I A_i is an ideal of 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_i 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 R_i. The ideal A is a prime ideal in R 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. However, the converse is not true when I is infinite. For example, the direct sum of the R_i form an ideal not contained in any such 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_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 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

• Direct product

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