# 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

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

considered as a subring of ${\mathbf{M}}_{2\mathrm{x}2}(R)$ (with standard matrix addition^{} and multiplication) is a zero ring. Moreover, the cardinality of this subset of ${\mathbf{M}}_{2\mathrm{x}2}(R)$ 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 $\prod _{j=1}^{m}}p({a}_{j})$, 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 |