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

considered as a subring of ${𝐌}_{2\mathrm{x}2}(R)$ (with standard matrix addition and multiplication) is a zero ring. Moreover, the cardinality of this subset of ${𝐌}_{2\mathrm{x}2}(R)$ 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 $p_1,\mathrm{\dots },p_m$ are distinct primes, $a_1,\mathrm{\dots },a_m$ are positive integers, and $n={\displaystyle \prod _{j=1}^m}p_j{}^{a_j}$, 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).