Prime ring

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 characteristic 0 field, the prime ring is the integers, for a characteristic "p" field (with "p" a prime number) the prime ring is the finite field of order "p" (cf. prime field).cite book |last=Lang |first=Serge |authorlink=Serge Lang |title=Algebra |edition=Third Edition|year=1997 |origyear=1965 |publisher=Addison-Wesley Publishing Company |location=USA |isbn=0-201-55540-9 |pages=p. 90]

Prime rings, under the first definition, can be regarded as a simultaneous generalization of both integral domains and matrix rings over fields.

Examples

* Any domain is a prime ring.
* Any simple ring is a prime ring, and more generally: every left or right primitive ring is a prime ring.
* Any matrix ring over an integral domain is a prime ring. In particular, the ring of 2-by-2 integer matrices is a prime ring.

Properties

* A commutative ring is a prime ring if and only if it is an integral domain.
* A ring is prime if and only if its zero ideal is a prime ideal.
* A non-trivial ring is prime if and only if the monoid of its ideals lacks zero divisors.
* The ring of matrices over a prime ring is again a prime ring.

References


Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Prime ideal — In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. This article only covers ideals of ring theory. Prime ideals in order theory are treated in the article on… …   Wikipedia

  • Prime number — Prime redirects here. For other uses, see Prime (disambiguation). A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is… …   Wikipedia

  • Prime Computer — Prime Computer, Inc. was a Natick, Massachusetts based producer of minicomputers from 1972 until 1992. The alternative spellings PR1ME and PR1ME Computer were used as brand names or logos by the company. Contents 1 Founders 2 History 3 Operating… …   Wikipedia

  • Prime — can refer to: * Prime number, an integer greater than 1 which is only divisible by 1 and itself * Prime (symbol), the ′ mark ** 3′ end and 5′ end ( three prime end , five prime end ) in biochemistry * Prime (liturgy), a liturgical office (service …   Wikipedia

  • 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

  • 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

  • Ring of mixed characteristic — In commutative algebra, a ring of mixed characteristic is a commutative ring R having characteristic zero and having an ideal I such that R / I has positive characteristic. Examples The integers Z have characteristic zero, but for any prime… …   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 Nebula (NGC 6822) — Diffuse nebula name = Ring Nebula in Barnard s Galaxy type = Emission epoch = J2000.0 ra = 19h 44m 36s dec = 14° 41 prime; 42 Prime; dist ly = 1.63 million ly dist pc = 0.500 Mpc appmag v = 14.5 size v = 53 Prime; constellation = Sagittarius… …   Wikipedia

  • Ring extension — In mathematics, more specifically in ring theory, a ring extension or extension ring is a ring R with a subring S . We write R / S and say R is a ring extension of S Given an extension R / S of commutative rings and a prime ideal P of R , it… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”