**Definition**. Let be a domain and If is left (resp. right) Ore, then is called a **left (resp. right) Ore domain***.*

**Remark 1. **Note that a domain is a left Ore domain if and only if for all non-zero elements Similarly is a right Ore domain if and only if for all non-zero elements

**Remark 2**. The left (resp. right) quotient ring of a left (resp. right) Ore domain is a division ring. The reason is that every element of is in the form where Now if then and then

**Theorem**. Every left (resp. right) Noetherian domain is left (resp. right) Ore.

*Proof.* We will prove the theorem for left Noetherian domains only. The “right” version, as usual, has the same argument. Let be non-zero. We need to show that So suppose, on the contrary, that We will show that the sum is direct and thus cannot be left Noetherian because then we would have the non-stopping increasing chain of left ideals

To prove that the sum is direct, suppose that the sum is not direct and choose to be the *smallest* postive integer for which there exist not all zero, such that Clearly Thus

Therefore, since are non-zero and is a domain, we must have and contradicting the minimality of