Product of rings

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 and are coprime integers, the quotient ring $\Z/mn\Z$ is the product of $\Z/m\Z$ and $\Z/n\Z.$

Examples

An important example is Z/''n''Z, the ring of integers modulo ''n''. If ''n'' is written as a product of prime powers (see Fundamental theorem of arithmetic),

:$n=p_1^ p_2^\cdots\ p_k^,$

where the ''p_i'' are distinct primes, then Z/''n''Z is naturally isomorphic to the product

:$\mathbf/p_1^\mathbf \ \times \ \mathbf/p_2^\mathbf \ \times \ \cdots \ \times \ \mathbf/p_k^\mathbf.$
This follows from the Chinese remainder theorem.

Properties

If is a product of rings, then for every ''i'' in ''I'' we have a surjective ring homomorphism which projects the product on the ''i'' th coordinate. The product ''R'' together with the projections ''p_i'' has the following universal property:
:if ''S'' is any ring and is a ring homomorphism for every ''i'' in ''I'', then there exists ''precisely one'' ring homomorphism such that for every ''i'' in ''I''.
This shows that the product of rings is an instance of products in the sense of category theory.

When ''I'' is finite, the underlying additive group of coincides with the direct sum of the additive groups of the ''R''_''i''. In this case, some authors call ''R'' the "direct sum of the rings ''R''_''i''" and write , 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''_''i'' are non-trivial, the inclusion map fails to map 1 to 1 and hence is not a ring homomorphism.

(A finite coproduct in the category 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.

If ''A_i'' is an ideal of ''R_i'' for each ''i'' in ''I'', then 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 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_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 the ''R_i''.

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''_''i'' (''x'') and ''y'' is an element of the product with all coordinates zero except ''p''_''j'' (''y'') where ''i'' ≠ ''j'', then ''xy'' = 0 in the product ring.

References

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{{DEFAULTSORT:Product Of Rings
Ring theory
Operations on structures