# zero ring

A ring is a 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 ZeroRing 2013-03-22 13:30:19 2013-03-22 13:30:19 Wkbj79 (1863) Wkbj79 (1863) 26 Wkbj79 (1863) Definition msc 16U99 msc 13M05 msc 13A99 ZeroVectorSpace Unity