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Concept
Product of rings
Summary
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 .
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
An important example is Z/nZ, the ring of integers modulo n. If n is written as a product of prime powers (see Fundamental theorem of arithmetic),
where the pi are distinct primes, then Z/nZ is naturally isomorphic to the product
This follows from the Chinese remainder theorem.
If R = Πi∈I Ri is a product of rings, then for every i in I we have a surjective ring homomorphism pi : R → Ri which projects the product on the i th coordinate. The product R together with the projections pi has the following universal property:
if S is any ring and fi : S → Ri is a ring homomorphism for every i in I, then there exists precisely one ring homomorphism f : S → R such that pi ∘ f = fi for every i in I.
This shows that the product of rings is an instance of .
When I is finite, the underlying additive group of Πi∈I Ri coincides with the direct sum of the additive groups of the Ri. In this case, some authors call R the "direct sum of the rings Ri" and write ⊕i∈I Ri, but this is incorrect from the point of view of , since it is usually not a coproduct in the (with identity): for example, when two or more of the Ri are non-trivial, the inclusion map Ri → R fails to map 1 to 1 and hence is not a ring homomorphism.
(A finite coproduct in the of commutative 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.
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