Encyclopedia > Product of rings

  Article Content

Product of rings

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.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Kings Park, New York

... of it is water. The total area is 6.21% water. Demographics As of the census of 2000, there are 16,146 people, 5,480 households, and 4,197 families residing in the ...

 
 
 
This page was created in 21.9 ms