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 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 resulting ring is called a direct product of the rings R_i. The direct product of finitely many rings R₁,...,R_k is also written as R₁ × R₂ × ... × R_k or R₁ ⊕ R₂ ⊕ ... ⊕ R_k, and can also be called the direct sum (and sometimes the complete direct sum^[1]) of the rings R_i.

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):

$n=p_1^{n_1}\ p_2^{n_2}\ \cdots\ p_k^{n_k}$

where the p_i are distinct primes, then Z/nZ is naturally isomorphic to the product ring

$\mathbf{Z}/p_1^{n_1}\mathbf{Z} \ \times \ \mathbf{Z}/p_2^{n_2}\mathbf{Z} \ \times \ \cdots \ \times \ \mathbf{Z}/p_k^{n_k}\mathbf{Z}$

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_i which projects the product on the i-th 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 o 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. 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_i to the identity in R.

If A_i in R_i is an ideal for each i in I, then A = Π_{i in 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 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

• Direct product

Notes

1. ^ Herstein 2005, p. 52

References

• Herstein, I.N. (2005) [1968], Noncommutative rings (5th ed.), Cambridge University Press, ISBN 978-0-883-85039-8 
• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, p. 91, ISBN 978-0-387-95385-4, MR1878556