 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 R_{1},...,R_{k} is also written as R_{1} × R_{2} × ... × R_{k} or R_{1} ⊕ R_{2} ⊕ ... ⊕ R_{k}, and can also be called the direct sum (and sometimes the complete direct sum^{[1]}) of the rings R_{i}.
Contents
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):
where the p_{i} are distinct primes, then Z/nZ 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_{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} 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 nonzero, 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 nonzero 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
 ^ Herstein 2005, p. 52
References
 Herstein, I.N. (2005) [1968], Noncommutative rings (5th ed.), Cambridge University Press, ISBN 9780883850398
 Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: SpringerVerlag, p. 91, ISBN 9780387953854, MR1878556
Categories: Ring theory
 Binary operations
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