Editing Product of rings

{{Short description|Ring built from other rings (mathematics)}}
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In [[mathematics]], a '''product of rings''' or '''direct product of rings''' is a [[ring (mathematics)|ring]] that is formed by the [[Cartesian product]] of the underlying sets of several rings (possibly an infinity), equipped with [[componentwise operation]]s. It is a [[direct product]] in the [[category of rings]].

Since direct products are defined [[up to]] an [[isomorphism]], one says colloquially that a ring is the product of some rings if it is isomorphic to the direct product of these rings. For example, the [[Chinese remainder theorem]] may be stated as: if {{mvar|m}} and {{mvar|n}} are [[coprime integers]], the [[quotient ring]] <math>\Z/mn\Z</math> is the product of <math>\Z/m\Z</math> and <math>\Z/n\Z.</math>

==Examples==
An important example is '''Z'''/''n'''''Z''', the [[ring of integers modulo n|ring of integers modulo ''n'']]. If ''n'' is written as a product of [[prime power]]s (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 [[prime number|primes]], then '''Z'''/''n'''''Z''' is naturally [[ring isomorphism|isomorphic]] to the product

:<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>''&thinsp;∘&thinsp;''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]]. 

When ''I'' is finite, the underlying additive group of {{nowrap|Π<sub>''i''∈''I''</sub> ''R''<sub>''i''</sub>}} coincides with the [[direct sum]] of the additive groups of the ''R''<sub>''i''</sub>. In this case, some authors call ''R'' the "direct sum of the rings ''R''<sub>''i''</sub>" and write {{nowrap|⊕<sub>''i''∈''I''</sub> ''R''<sub>''i''</sub>}}, but this is incorrect from the point of view of [[category theory]], since it is usually not a [[coproduct]] in the [[category of rings]] (with identity): for example, when two or more of the ''R''<sub>''i''</sub> are non-[[trivial ring|trivial]], the inclusion map {{nowrap|''R<sub>i</sub>'' → ''R''}} fails to map 1 to 1 and hence is not a ring homomorphism.

(A finite coproduct in the [[category (mathematics)|category]] of [[commutative algebra (structure)|commutative]] [[algebra over a commutative ring|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]] natural isomorphism, meaning that it doesn't matter in which order one forms the direct product.

If ''A<sub>i</sub>'' is an [[ideal (ring theory)|ideal]] of ''R<sub>i</sub>'' 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 (logic)|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-trivial, 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 (ring theory)|unit]] [[if and only if]] all of its components are units, i.e., if and only if ''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 the ''R<sub>i</sub>''.

A product of two or more non-trivial rings always has nonzero [[zero divisors]]: if ''x'' is an element of the product whose coordinates are  all zero except ''p''<sub>''i''</sub>(''x'') and ''y'' is an element of the product with all coordinates zero except ''p''<sub>''j''</sub>(''y'') where ''i''&nbsp;≠&nbsp;''j'', then ''xy''&nbsp;=&nbsp;0 in the product ring.

==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
| orig-year=1968
| url-access=registration
| url=https://archive.org/details/noncommutativeri0000hers
}}
*{{Lang Algebra|edition=3r|page=91}}

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[[Category:Ring theory]]
[[Category:Operations on structures]]