Product of rings

From Academic Kids

fr:Produit d'anneaux In abstract algebra, 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 in I Ri can be turned into a ring by defining the operations coordinatewise, i.e.

(ai) + (bi) = (ai + bi)
(ai) (bi) = (ai bi)

The product of finitely many rings R1,...,Rk is also written as R1 × R2 × ... × Rk.

Examples

The most 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):

<math>n={p_1}^{n_1}\ {p_2}^{n_2}\ \cdots\ {p_k}^{n_k}<math>

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

<math>\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}<math>

This follows from the Chinese remainder theorem.

Properties

If R = Πi in I Ri 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 i-th coordinate. The product R, together with the projections pi, has the following universal property:

if S is any ring and fi : S -> Ri is a ring homomorphism for every i in I, then there exists precisely one ring homomorphism f : S -> R such that pi o f = fi for every i in I.

This shows that the product of rings is an instance of products in the sense of category theory.

If A is a (left, right, two-sided) ideal in R, then there exist (left, right, two-sided) ideals Ai in Ri such that A = Πi in I Ai. Conversely, every such product of ideals is an ideal in R. A is a prime ideal in R if and only if all but one of the Ai are equal to Ri and the remaining Ai is a prime ideal in Ri.

An element x in R is a unit if and only if all of its components are units, i.e. if and only if pi(x) is a unit in Ri for every i in I. The group of units of R is the product of the groups of units of Ri.

A product of more than one non-zero rings always has zero divisors: if x is an element of the product all of whose coordinates are zero except pi(x), and y is an element of the product with all coordinates zero except pj(y) (with ij), then xy = 0 in the product ring.

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