Goldie ring

Summary

A right Goldie ring is a ring with two finiteness conditions. It contains no infinite direct sum of nonzero right ideals, and it has the ascending chain condition on right annihilators. Every right Noetherian ring is right Goldie, and so is every commutative integral domain. These are the conditions in Goldie's theorem, which describes the semiprime right Goldie rings as the rings with a semisimple Artinian classical ring of quotients.