zero ring ZeroRing 2003-03-10 03:07:48 2007-06-12 19:18:17 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here A ring is a {\sl 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 \PMlinkname{isomorphism}{RingIsomorphism}, this is the only zero ring that has a multiplicative identity. Zero rings exist in \PMlinkescapetext{abundance}. 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 \PMlinkname{fundamental theorem of finite abelian groups}{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 \PMlinkname{order}{Order} $n$ is $\displaystyle \prod_{j=1}^m p(a_j)$, where $p$ denotes the \PMlinkname{partition function}{PartitionFunction2}.