I was reading my lecture notes and the symbol $/$ is used commonly but it is not really defined anywhere, so I was wondering if someone could explain it to me.
For example, in the definition of a quotient ring from an ideal
$$R/I = \{ a + I : a \in R \}$$
where $I$ is an ideal and $R$ is a ring. What exactly does the forward slash mean?
The equation $$R/I = \{ a + I : a \in R \}$$ is the definition of the notation $R/I$. So you're not expected to know what the forward slash means before this definition: this is telling you what it means!