In this post, is a unitary ring, is multiplicatively closed with and

**Definition.** is called a **left Ore** set if for all Similarly is called a **right Ore** set if for all

**Remark 1**. If is the left (resp., right) quotient ring of with respect to then is a left (resp., right) Ore set. The reason is that if is the corresponding map, then for any we must have for some Thus and hence So for some Hence

**Remark 2**. If is a left Ore set, then is an ideal of (Recall that if has the left quotient ring , then is nothing but the kernel of the corresponding map from to ). To see this, we note that is clearly closed under right multiplication by elements of also:

i) for every To prove this, we have for some By Remark 1, Therefore for some So and hence i.e.

ii) for every To prove this one, we have for some Now, by Remark 1: So for some Thus i.e.

iii) if is a right Ore set, then

**Remark 3**. Suppose is left Ore and There exist such that The proof is by induction over : if then we may choose Suppose and the claim is true for Choose so that Also there exist and such that since Let for Then for all we have and Similarly, if is right Ore and then there exist such that