zero ring

A ring is a zero ringMathworldPlanetmath if the product of any two elements is the additive identity (or zero).

Zero rings are commutativePlanetmathPlanetmathPlanetmath under multiplication. For if Z is a zero ring, 0Z is its additive identity, and x,yZ, then xy=0Z=yx.

Every zero ring is a nilpotent ring. For if Z is a zero ring, then Z2={0Z}.

Since every subring of a ring must contain its zero elementMathworldPlanetmath, every subring of a ring is an ideal, and a zero ring has no prime idealsMathworldPlanetmathPlanetmath.

The simplest zero ring is 1={0}. Up to isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (, this is the only zero ring that has a multiplicative identityPlanetmathPlanetmath.

Zero rings exist in . They can be constructed from any ring. If R is a ring, then


considered as a subring of 𝐌2x2(R) (with standard matrix additionMathworldPlanetmath and multiplication) is a zero ring. Moreover, the cardinality of this subset of 𝐌2x2(R) is the same as that of R.

Moreover, zero rings can be constructed from any abelian group. If G is a group with identityPlanetmathPlanetmathPlanetmath eG, it can be made into a zero ring by declaring its addition to be its group operation and defining its multiplication by ab=eG for any a,bG.

Every finite zero ring can be written as a direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmath of cyclic rings, which must also be zero rings themselves. This follows from the fundamental theorem of finite abelian groups ( Thus, if p1,,pm are distinct primes, a1,,am are positive integers, and n=j=1mpjaj, then the number of zero rings of order ( n is j=1mp(aj), where p denotes the partition function (

Title zero ring
Canonical name ZeroRing
Date of creation 2013-03-22 13:30:19
Last modified on 2013-03-22 13:30:19
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 26
Author Wkbj79 (1863)
Entry type Definition
Classification msc 16U99
Classification msc 13M05
Classification msc 13A99
Related topic ZeroVectorSpace
Related topic Unity