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 ${\displaystyle \mathbb {Z} /mn\mathbb {Z} }$ is the product of ${\displaystyle \mathbb {Z} /m\mathbb {Z} }$ and ${\displaystyle \mathbb {Z} /n\mathbb {Z} .}$

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 p_i are distinct primes, then Z/nZ is naturally isomorphic to the product

This follows from the Chinese remainder theorem.

If R = Π_i∈I R_i is a product of rings, then for every i in I we have a surjective ring homomorphism p_i : R → R_i which projects the product on the ith coordinate. The product R together with the projections p_i 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 Π_i∈I R_i 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 ⊕_i∈I R_i, 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 R_i → R fails to map 1 to 1 and hence is not a ring homomorphism.