# Trivial ring

﻿
Trivial ring

A trivial ring is a ring defined on a singleton set, {"r"}. The ring operations (&times; and +) are trivial:

:$r imes r = r$:$r + r = r ,$ Clearly this ring is commutative. Its single element is both the additive and the multiplicative identity element, i.e., :$r = 0 = 1 ,$

A ring "R" is trivial if and only if 1 = 0, since this equality implies that for all "r" within "R", :$r = r imes 1 = r imes 0 = 0$

Any two trivial rings are isomorphic, so one may also speak about "the" trivial ring.

The trivial ring is also sometimes called the zero ring, because {0} (where 0 here corresponds to the number 0) is a ring under the standard operations of addition and multiplication.

References

*

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Ring (mathematics) — This article is about algebraic structures. For geometric rings, see Annulus (mathematics). For the set theory concept, see Ring of sets. Polynomials, represented here by curves, form a ring under addition and multiplication. In mathematics, a… …   Wikipedia

• Trivial (mathematics) — In mathematics, the term trivial is frequently used for objects (for examples, groups or topological spaces) that have a very simple structure. For non mathematicians, they are sometimes more difficult to visualize or understand than other, more… …   Wikipedia

• Ring theory — In abstract algebra, ring theory is the study of rings algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. Ring theory studies the structure of rings, their… …   Wikipedia

• Ring circuit — In electricity supply, a ring final circuit or ring circuit (informally also ring main or just ring) is an electrical wiring technique developed and primarily used in the United Kingdom that provides two independent conductors for live, neutral… …   Wikipedia

• Trivial File Transfer Protocol — TFTP (Trivial File Transfer Protocol) Familie: TCP/IP Einsatzgebiet: Laden von Betriebssystemen und anderen Daten über das Netzwerk Ports: 69/UDP TFTP im TCP/IP‑Protokollstapel: Anwendung TFTP Transport …   Deutsch Wikipedia

• Trivial FTP — TFTP Название: Trivial File Transfer Protocol Уровень (по модели OSI): Прикладной Семейство: IP Создан в: 1980 г. Порт/ID: 69/UDP Назначение протокола: Передача файлов Спецификация: RFC 1350 / …   Википедия

• Trivial File Transfer Protocol — TFTP Название: Trivial File Transfer Protocol Уровень (по модели OSI): Прикладной Семейство: IP Создан в: 1980 г. Порт/ID: 69/UDP Назначение протокола: Передача файлов Спецификация: RFC 1350 / …   Википедия

• Glossary of ring theory — Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject. Contents 1 Definition of a ring 2 Types of… …   Wikipedia

• Quotient ring — In mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in ring theory, quite similar to the factor groups of group theory and the quotient spaces of linear algebra. One starts with a ring R and a two… …   Wikipedia

• Prime ring — In abstract algebra, a non trivial ring R is a prime ring if for any two elements a and b of R , if arb = 0 for all r in R , then either a = 0 or b = 0 . Prime ring can also refer to the subring of a field determined by its characteristic. For a… …   Wikipedia