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In mathematics, a product of rings or direct product of rings is a ring that is formed by the Cartesian product of the underlying sets of several rings (possibly an infinity), equipped with componentwise operations. 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 m and n are coprime integers, the quotient ring is the product of and

Attributes	Values
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rdfs:label	Produit d'anneaux (fr) 環の直積 (ja) Product van ringen (nl) Product of rings (en)
rdfs:comment	En algèbre générale, il est possible de combiner plusieurs anneaux pour former un anneau appelé anneau produit. (fr) In mathematics, a product of rings or direct product of rings is a ring that is formed by the Cartesian product of the underlying sets of several rings (possibly an infinity), equipped with componentwise operations. 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 m and n are coprime integers, the quotient ring is the product of and (en) 数学において、いくつかの環を1つの大きい直積環、積環 (product ring) に合併することができる。これは次のようにされる： I がある添え字集合で Ri が I のすべての i に対して環であれば、カルテジアン積 Πi ∈ I Ri は演算を成分ごとの演算として定義することによって環にできる。得られる環は環 Ri の直積 (direct product) と呼ばれる。有限個の環の直積は環の直和 (direct sum) と一致する。 (ja) In de ringtheorie, een deelgebied van de wiskunde, is het mogelijk om verschillende ringen te combineren tot een grotere productring. Het directe product van de samenstellende ringen. Het directe product van de ringen , met een willekeurige indexverzameling wordt gevormd door het cartesisch product met als bewerkingen de coördinaatsgewijze uitgevoerde bewerkingen van de samenstellende ringen. Dat houdt in dat voor de elementen en geldt: en (nl)
dct:subject	Ring theory Operations on structures
Wikipage page ID	25063 (xsd:integer)
Wikipage revision ID	1083592571 (xsd:integer)
Link from a Wikipage to another Wikipage	Cambridge University Press Cartesian product Prime ideal Prime power Ring isomorphism Unit (ring theory) Quotient ring Ring theory Mathematics Fundamental theorem of arithmetic Converse (logic) Coprime integers Coproduct Ideal (ring theory) Maximal ideal A fortiori Operations on structures Direct product Direct sum Direct sum of modules Isomorphism Product (category theory) Ring (mathematics) Prime number Surjective Chinese remainder theorem Tensor product of algebras Zero divisors Direct product of groups Axiom of choice Group of units If and only if Algebra over a commutative ring Category (mathematics) Category of rings Category theory Up to Universal property Ring homomorphism Free product of algebras Ring of integers modulo n Commutative algebra (structure) Componentwise operation Trivial ring
Link from a Wikipage to an external page	https://archive.org/details/noncommutativeri0000hers
sameAs	Product of rings Product of rings Product of rings Product of rings Product of rings Product of rings Product of rings
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Mvar dbt:Ring_theory_sidebar dbt:Short_description dbt:Lang_Algebra
page	91 (xsd:integer)
edition	3 (xsd:integer)
has abstract	En algèbre générale, il est possible de combiner plusieurs anneaux pour former un anneau appelé anneau produit. (fr) In mathematics, a product of rings or direct product of rings is a ring that is formed by the Cartesian product of the underlying sets of several rings (possibly an infinity), equipped with componentwise operations. 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 m and n are coprime integers, the quotient ring is the product of and (en) 数学において、いくつかの環を1つの大きい直積環、積環 (product ring) に合併することができる。これは次のようにされる： I がある添え字集合で Ri が I のすべての i に対して環であれば、カルテジアン積 Πi ∈ I Ri は演算を成分ごとの演算として定義することによって環にできる。得られる環は環 Ri の直積 (direct product) と呼ばれる。有限個の環の直積は環の直和 (direct sum) と一致する。 (ja) In de ringtheorie, een deelgebied van de wiskunde, is het mogelijk om verschillende ringen te combineren tot een grotere productring. Het directe product van de samenstellende ringen. Het directe product van de ringen , met een willekeurige indexverzameling wordt gevormd door het cartesisch product met als bewerkingen de coördinaatsgewijze uitgevoerde bewerkingen van de samenstellende ringen. Dat houdt in dat voor de elementen en geldt: en (nl)
prov:wasDerivedFrom	wikipedia-en:Product_of_rings?oldid=1083592571&ns=0
page length (characters) of wiki page	5738 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Product_of_rings
is Link from a Wikipage to another Wikipage of	Product ring List of abstract algebra topics List of commutative algebra topics Quotient ring Zero divisor Frobenius algebra Idempotent (ring theory) Matrix ring Wedderburn–Artin theorem Noncommutative ring Fourier transform on finite groups Product Product (mathematics) Chinese remainder theorem Bimodule

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