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, this is the only zero ring that has a multiplicative identity.

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

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. 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 $n$ is $\displaystyle \prod_{j=1}^m p(a_j)$ , where $p$ denotes the partition function.