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{{refimprove|date=November 2012}}
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In [[mathematics]], it is possible to combine several [[ring (mathematics)|rings]] into one large '''product ring'''. This is done as follows: if ''I'' is some [[index set]] and ''R<sub>i</sub>'' is a ring for every ''i'' in ''I'', then the [[cartesian product]] {{nowrap|Π<sub>''i'' ∈ ''I''</sub> ''R''<sub>''i''</sub>}} can be turned into a ring by defining the operations coordinate-wise.
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The resulting ring is called a '''direct product''' of the rings ''R''<sub>''i''</sub>. 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 [[integer]]s [[modular arithmetic|modulo]] ''n''. If ''n'' is written as a product of [[prime number|prime]] powers (see [[fundamental theorem of arithmetic]]):
 
:<math>n=p_1^{n_1}\  p_2^{n_2}\ \cdots\ p_k^{n_k}</math>
 
where the ''p<sub>i</sub>'' are distinct primes, then '''Z'''/''n'''''Z''' is naturally [[isomorphic]] to the product ring
 
:<math>\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 {{nowrap|1=''R'' = Π<sub>''i'' ∈ ''I''</sub> ''R''<sub>''i''</sub>}} is a product of rings, then for every ''i'' in ''I'' we have a [[surjective]] [[ring homomorphism]] {{nowrap|''p<sub>i</sub>'': ''R'' → ''R<sub>i</sub>''}} which projects the product on the ''i''th coordinate. The product ''R'', together with the projections ''p<sub>i</sub>'', has the following [[universal property]]:
 
:if ''S'' is any ring and {{nowrap|''f<sub>i</sub>'': ''S'' → ''R<sub>i</sub>''}} is a ring homomorphism for every ''i'' in ''I'', then there exists ''precisely one'' ring homomorphism {{nowrap|''f'': ''S'' → ''R''}} such that {{nowrap|1=''p<sub>i</sub>'' ∘ ''f'' = ''f<sub>i</sub>''}} for every ''i'' in ''I''.
 
This shows that the product of rings is an instance of [[product (category theory)|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 {{nowrap|''R<sub>i''</sub> → ''R''}} is not ring homomorphism as it does not map the identity in ''R<sub>i''</sub> to the identity in ''R''.
 
If ''A<sub>i</sub>'' in ''R<sub>i</sub>'' is an [[ideal (ring theory)|ideal]] for each ''i'' in ''I'', then {{nowrap|1=''A'' = Π<sub>''i'' ∈ ''I''</sub> ''A<sub>i</sub>''}} 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<sub>i</sub>'' 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<sub>i</sub>''.  The ideal ''A'' is a [[prime ideal]] in ''R'' if all but one of the ''A<sub>i</sub>'' are equal to ''R<sub>i</sub>'' and the remaining ''A<sub>i</sub>'' is a prime ideal in ''R<sub>i</sub>''. However, the converse is not true when ''I'' is infinite. For example, the [[Direct sum of modules|direct sum]] of the ''R<sub>i</sub>'' 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 {{nowrap|''p<sub>i</sub>''(''x'')}} is a unit in ''R<sub>i</sub>'' for every ''i'' in ''I''. The group of units of ''R'' is the [[direct product of groups|product]] of the groups of units of ''R<sub>i</sub>''.
 
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 {{nowrap|''p<sub>i</sub>''(''x'')}}, and ''y'' is an element of the product with all coordinates zero except {{nowrap|''p<sub>j</sub>''(''y'')}} (with {{nowrap|''i'' ≠ ''j''}}), then {{nowrap|1=''xy'' = 0}} in the product ring.
 
==See also==
*[[Direct product]]
 
==Notes==
{{reflist}}
 
==References==
*{{Citation
| last=Herstein
| first=I.N.
| author-link=Israel Nathan Herstein
| title=Noncommutative rings
| year=2005
| publisher=[[Cambridge University Press]]
| edition=5th
| isbn=978-0-88385-039-8
| origyear=1968
}}
*{{Lang Algebra|edition=3r|page=91}}
 
{{DEFAULTSORT:Product Of Rings}}
[[Category:Ring theory]]
[[Category:Binary operations]]

Revision as of 11:53, 5 March 2014

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