Goldie’s theorem

Let R be a ring with an identityPlanetmathPlanetmath. Then R has a right classical ring of quotients Q which is semisimplePlanetmathPlanetmathPlanetmathPlanetmath ArtinianPlanetmathPlanetmath if and only if R is a semiprime right Goldie ring. If this is the case, then the compositionMathworldPlanetmathPlanetmath length of Q is equal to the uniform dimension of R.

An immediate corollary of this is that a semiprime right Noetherian ring always has a right classical ring of quotients.

This result was discovered by Alfred in the late 1950’s.

Title Goldie’s theorem
Canonical name GoldiesTheorem
Date of creation 2013-03-22 14:04:17
Last modified on 2013-03-22 14:04:17
Owner mclase (549)
Last modified by mclase (549)
Numerical id 7
Author mclase (549)
Entry type Theorem
Classification msc 16U20
Classification msc 16P60
Related topic OresTheorem2