In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Goldie's theorem is a basic structural result in
ring theory, proved by
Alfred Goldie during the 1950s. What is now termed a right Goldie ring is a
ring
(The) Ring(s) 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
Arts, entertainment, and media Film and TV
* ''The Ring'' (franchise), a ...
''R'' that has finite
uniform dimension (="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 Ideal (ring theory), ideals in certain commutative rings. These conditions p ...
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 n ...
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. 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-Noethe ...
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 is a right Goldie domain, and hence so is every commutative
integral domain
In mathematics, 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 setting for studying divisibilit ...
.
A consequence of Goldie's theorem, again due to Goldie, is that every semiprime
principal right ideal ring 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 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 ideals in ''R''.
** ''R'' is a right
nonsingular ring.
[This may be deduced from a theorem of Mewborn and Winton, that if a ring satisfies the maximal condition on right annihilators then the right singular ideal is nilpotent. ]
** From the previous observations, ''R'' is a right
Ore ring, and so its right classical ring of quotients ''Q''
''r'' exists. Also from the previous observations, ''Q''
''r'' is a
semisimple ring
In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itself ...
. 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
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External links
PlanetMath page on Goldie's theorem
Theorems in ring theory
{{abstract-algebra-stub