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 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 ${\displaystyle \mathbb {Z} /n\mathbb {Z} .}$

Examples

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),

${\displaystyle n=p_{1}^{n_{1}}p_{2}^{n_{2}}\cdots \ p_{k}^{n_{k}},}$

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

${\displaystyle \mathbf {Z} /p_{1}^{n_{1}}\mathbf {Z} \ \times \ \mathbf {Z} /p_{2}^{n_{2}}\mathbf {Z} \ \times \ \cdots \ \times \ \mathbf {Z} /p_{k}^{n_{k}}\mathbf {Z} .}$

This follows from the Chinese remainder theorem.

Properties

If R = Π_i∈IR_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 i th coordinate. The product R together with the projections p_i has the following universal property:

if S is any ring and f_i : S → R_i is a ring homomorphism for every i in I, then there exists precisely one ring homomorphism f : S → R such that p_i ∘ f = f_i 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 Π_i∈IR_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∈IR_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.

(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 A = Π_i∈IA_i 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

• Herstein, I.N. (2005) [1968], Noncommutative rings (5th ed.), Cambridge University Press, ISBN   978-0-88385-039-8
• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, p. 91, ISBN   978-0-387-95385-4, MR   1878556, Zbl   0984.00001