Let $R$ be a ring. If the set of annihilators $\{\rann(x)\mid x\in R\}$ 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.