Goldie ring

Let $R$ be a ring. If the set of annihilators $\{\mathrm{r}.\mathrm{ann}(x)\mid x\in R\}$ satisifies the ascending chain condition, then $R$ is said to satisfy the ascending chain condition on right annihilators.

A ring $R$ is called a right Goldie ring if it satisfies the ascending chain condition on right annihilators and $R_R$ is a module of finite rank.

Left Goldie ring is defined similarly. If the context makes it clear on which side the ring operates, then such a ring is simply called a Goldie ring.

A right Noetherian ring is right Goldie.

Title	Goldie ring
Canonical name	GoldieRing
Date of creation	2013-03-22 14:04:10
Last modified on	2013-03-22 14:04:10
Owner	mclase (549)
Last modified by	mclase (549)
Numerical id	6
Author	mclase (549)
Entry type	Definition
Classification	msc 16P60
Synonym	Goldie
Related topic	UniformDimension
Defines	left Goldie
Defines	right Goldie
Defines	left Goldie ring
Defines	right Goldie ring