mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Goldie's theorem is a basic structural result in

ring theory In algebra, ring theory is the study of rings— algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their re ...

, proved by

Alfred Goldie Alfred William Goldie (10 December 1920, Coseley, Staffordshire – 8 October 2005, Barrow-in-Furness, Cumbria) was an English mathematician. Biography Goldie was educated at Wolverhampton Grammar School and then read mathematics at St John's Co ...

during the 1950s. What is now termed a right Goldie ring is a

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

''R'' that has finite

uniform dimension In abstract algebra, a module is called a uniform module if the intersection of any two nonzero submodules is nonzero. This is equivalent to saying that every nonzero submodule of ''M'' is an essential submodule. A ring may be called a right (left ...

(="finite rank") as a right module over itself, and satisfies the

ascending chain condition In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings.Jacobson (2009), p. 142 and 147 These con ...

on right annihilators of subsets of ''R''. Goldie's theorem states that the

semiprime In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there are infinitely many prime nu ...

right Goldie rings are precisely those that have a

semisimple In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...

Artinian right

classical ring of quotients In mathematics, especially in the area of algebra known as ring theory, the Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, o ...

. The structure of this ring of quotients is then completely determined by the Artin–Wedderburn theorem. In particular, Goldie's theorem applies to semiprime right

Noetherian ring In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noether ...

s, since by definition right Noetherian rings have the ascending chain condition on ''all'' right ideals. This is sufficient to guarantee that a right-Noetherian ring is right Goldie. The converse does not hold: every right

Ore domain In mathematics, especially in the area of algebra known as ring theory, the Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, ...

is a right Goldie domain, and hence so is every commutative

integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural set ...

. A consequence of Goldie's theorem, again due to Goldie, is that every semiprime

principal right ideal ring In mathematics, a principal right (left) ideal ring is a ring ''R'' in which every right (left) ideal is of the form ''xR'' (''Rx'') for some element ''x'' of ''R''. (The right and left ideals of this form, generated by one element, are called prin ...

is isomorphic to a finite direct sum of

prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

principal right ideal rings. Every prime principal right ideal ring is isomorphic to a

matrix ring In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ''U ...

over a right Ore domain.

Sketch of the proof

This is a sketch of the characterization mentioned in the introduction. It may be found in . *If ''R'' be a semiprime right Goldie ring, then it is a right order in a semisimple ring: ** Essential right ideals of ''R'' are exactly those containing a regular element. ** There are no non-zero

nil ideal In mathematics, more specifically ring theory, a left, right or two-sided ideal of a ring is said to be a nil ideal if each of its elements is nilpotent., p. 194 The nilradical of a commutative ring is an example of a nil ideal; in fact, it is ...

s in ''R''. ** ''R'' is a right

nonsingular ring In the branches of abstract algebra known as ring theory and module theory, each right (resp. left) ''R''-module ''M'' has a singular submodule consisting of elements whose annihilators are essential right (resp. left) ideals in ''R''. In set no ...

.This may be deduced from a theorem of Mewborn and Winton, that if a ring satisfies the

maximal condition In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings.Jacobson (2009), p. 142 and 147 These con ...

on right annihilators then the right singular ideal is nilpotent. ** From the previous observations, ''R'' is a right

Ore ring In mathematics, especially in the area of algebra known as ring theory, the Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, o ...

, and so its right classical ring of quotients ''Q''^''r'' exists. Also from the previous observations, ''Q''^''r'' is a semisimple ring. Thus ''R'' is a right order in ''Q''^''r''. * If ''R'' is a right order in a semisimple ring ''Q'', then it is semiprime right Goldie: **Any right order in a Noetherian ring (such as ''Q'') is right Goldie. **Any right order in a Noetherian semiprime ring (such as ''Q'') is itself semiprime. **Thus, ''R'' is semiprime right Goldie.

References

* * * * *

External links

PlanetMath page on Goldie's theorem

Theorems in ring theory {{abstract-algebra-stub