Throughout this section is a ring, unitary as usual, and a multiplicatively closed subset of such that

**Definition 1.** A ring is said to be a **left quotient ring** of with respect to if there exists a ring homomorphism such that the following conditions are satisfied:

1) is a unit in for all

2) every element of is in the form for some

3)

**Remark**. If for some then for some This is because

and thus since, by the condition 1), is a unit of

**Definition 2**. A ring is said to be a **right quotient ring** of (with respect to ) if there exists a ring homomorphism such that the following conditions are satisfied:

1) is a unit in for all

2) every element of is in the form for some

3)

**Lemma**. Suppose is a ring homomorphism and is a left (resp. right) quotient ring of with respect to If is a unit in for every then there exists a (unique) ring homomorphism which extends

*Poof.* If is a left quotient ring of and is the map in Definition 1, then for all we define

Similarly, if is a right quotient ring, then we define for all and Proving that is a well-defined ring homomorphism is lengthy. I will only prove that is well-defined:

Suppose for some and Then

which gives us for some and which satisfy

So and Hence for some . Therefore

and ,

which will give us

**Theorem**. A left (resp. right) quotient ring with respect to , if it exists, is unique up to isomorphism. If has a left quotient ring and a right quotient ring (with respect to ), then

*Proof.* An easy consequence of the lemma.

**Notation**. “The” left (resp. right) quotient ring of w.r.t. , if it exists, is called the left (resp. right) localization of at and it is denoted by (resp. .