Great description. I’d say it’s also important to note that in this context, the division slash means you divide the “numerator” into equivalence classes such that the difference between any two elements in a class is divisible by the “denominator”; i.e., class elements are equivalent mod the denominator. So for instance, x^2 and -1 lie in the same class because x^2-(-1) = x^2 + 1
Yeah, that's important for quotients by more general ideals, but for a principal ideal in a polynomial ring, it's easier to speak of setting the generator to zero.
