reduced ring

A ring R is said to be a reduced ring if R contains no non-zero nilpotent elementsMathworldPlanetmath. In other words, r2=0 implies r=0 for any rR.

Below are some examples of reduced rings.

  • A reduced ring is semiprimePlanetmathPlanetmath.

  • A ring is a domain ( iff it is prime ( and reduced.

  • A commutativePlanetmathPlanetmathPlanetmath semiprime ring is reduced. In particular, all integral domainsMathworldPlanetmath and Boolean ringsMathworldPlanetmath are reduced.

  • Assume that R is commutative, and let A be the set of all nilpotent elements. Then A is an ideal of R, and that R/A is reduced (for if (r+A)2=0, then r2A, so r2, and consequently r, is nilpotentPlanetmathPlanetmath, or rA).

An example of a reduced ring with zero-divisors is n, with multiplicationPlanetmathPlanetmath defined componentwise: (a1,,an)(b1,,bn):=(a1b1,,anbn). A ring of functions taking values in a reduced ring is also reduced.

Some prototypical examples of rings that are not reduced are 8, since 42=0, as well as any matrix ring over any ring; as illustrated by the instance below

Title reduced ring
Canonical name ReducedRing
Date of creation 2013-03-22 14:18:12
Last modified on 2013-03-22 14:18:12
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 17
Author CWoo (3771)
Entry type Definition
Classification msc 16N60
Synonym nilpotent-free