After Semigroup and Monoid, the rest of PureScript's Numeric type classes (e.g.
Ring, etc.) and all the mathematical notations they use are very clearly explained elsewhere in hdgarrood's Numeric Hierarchy Overview.
Another resource that might be helpful is A Brief Guide to A Few Algebraic Data Structures. However, this is more of a reference material than a tutorial like Harry's above overview.
For derived functions (if any), see the type classes' docs.