zero ring

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

Zero rings are commutative under multiplication. For if $Z$ is a zero ring, $0_{Z}$ is its additive identity, and $x,y\in Z$ , then $xy=0_{Z}=yx.$

Every zero ring is a nilpotent ring. For if $Z$ is a zero ring, then $Z^{2}=\{0_{Z}\}$ .

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

The simplest zero ring is ${\mathbb{Z}}_{1}=\{0\}$ . Up to isomorphism (http://planetmath.org/RingIsomorphism), this is the only zero ring that has a multiplicative identity.

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

$\left\{\left.\left(\begin{array}[]{cc}r&-r\\ r&-r\end{array}\right)\right|r\in R\right\}$

considered as a subring of ${\mathbf{M}}_{2\operatorname{x}2}(R)$ (with standard matrix addition and multiplication) is a zero ring. Moreover, the cardinality of this subset of ${\mathbf{M}}_{2\operatorname{x}2}(R)$ is the same as that of $R$ .

Moreover, zero rings can be constructed from any abelian group. If $G$ is a group with identity $e_{G}$ , it can be made into a zero ring by declaring its addition to be its group operation and defining its multiplication by $a\cdot b=e_{G}$ for any $a,b\in G$ .

Every finite zero ring can be written as a direct product of cyclic rings, which must also be zero rings themselves. This follows from the fundamental theorem of finite abelian groups (http://planetmath.org/FundamentalTheoremOfFinitelyGeneratedAbelianGroups). Thus, if $p_{1},\ldots,p_{m}$ are distinct primes, $a_{1},\ldots,a_{m}$ are positive integers, and $\displaystyle n=\prod_{j=1}^{m}{p_{j}}^{a_{j}}$ , then the number of zero rings of order (http://planetmath.org/Order) $n$ is $\displaystyle\prod_{j=1}^{m}p(a_{j})$ , where $p$ denotes the partition function (http://planetmath.org/PartitionFunction2).

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