Product of rings
In mathematics, it is possible to combine several rings into one large product ring. This is done as follows: if I is some index set and Ri is a ring for every i in I, then the cartesian product Πi ∈ IRi can be turned into a ring by defining the operations coordinate-wise.
The resulting ring is called a direct product of the rings Ri.
An important example is the ring Z/nZ 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 ring
This follows from the Chinese remainder theorem.
If R = Πi ∈ IRi 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 ith coordinate. The product R, together with the projections pi, has the following universal property:
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 Πi ∈ IRi 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 ∈ IRi, but this is incorrect from the point of view of category theory, since it is usually not a coproduct in the category of rings: for example, when two or more of the Ri are nonzero, the inclusion map Ri → R fails to map 1 to 1 and hence is not a ring homomorphism.
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