If $A$ is a ring and $S$ a multiplicative set, how does the elements of $S^{-1}A$ look like? In my book one introduces the equivalence relation $\sim$ on $A \times S$ as follows: $(a,s) \sim (b,t) \Longleftrightarrow \exists u \in S$ $\text{such that}$ $u(at-bs)=0$
Then one states that $\sim$ is an equivalence relation (this is ok), however I dont understand this statement: "Then the ring of fractions of $A$ with respect to $S$ is $S^{-1}A=(A \times S)/ \sim$"
What does the ring of fractions of $A$ with respect to $S$ mean? I understand that $(A \times S)/ \sim$ consists of equivalence classes under the relation $\sim$, but what has this to do with $S^{-1}A$ (a ring of fractions)?
It is a new ring built from $A$ and a subset $S$ which is well-behaved enough to act like denominators of fractions. The elements of $S$ become units in the new ring, which they may not have been in the old ring.
The main idea is that each fraction is an equivalence class in that relation. The idea is to think of $(a,s)$ as $\frac{a}{s}$.
You are literally defining the set of fractions in terms of A and S. What else is there to say?