Algebraic structure → Ring theory Ring theory |
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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

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),

where the *p _{i}* are distinct primes, then

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}* :

- if
*S*is any ring and*f*:_{i}*S*→*R*is a ring homomorphism for every_{i}*i*in*I*, then there exists*precisely one*ring homomorphism*f*:*S*→*R*such that*p*∘_{i}*f*=*f*for every_{i}*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 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}* →

(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

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

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.