How many elements does $\mathbb{Z}[1/6]/(2020)\mathbb{Z}[1/6]$ have?

Question

Consider the $\mathbb{Z}$-module $\mathbb{Z}[1/6]$ and the module $(2020)\subset \mathbb{Z}$. In an algebra exercise from last year, it was asked how many elements the quotient $\mathbb{Z}[1/6]/(2020)\mathbb{Z}[1/6]$ has (where $(2020)\mathbb{Z}[1/6]$ is the submodule of $\mathbb{Z}[1/6]$ as expected). However, it seems to me that it is infinitely large, as $\mathbb{Z}[1/6]$ contains infinitely many elements (1,1/6,1/36,... as example, and I do not see how these elements differ in the quotient either). Does anyone know how to solve this problem?

Answer 1

In general, $$S^{-1}A/IS^{-1}A\simeq T^{-1}(A/I),$$ where $T$ is the image of $S$ in $A/I$.

In your case, the quotient ring is isomorphic to the ring of fractions of $\mathbb Z/2020\mathbb Z$ with respect to the multiplicative set $T$ of the powers of $\hat 6$. At its turn, this ring is isomorphic to a quotient ring of $\mathbb Z/2020\mathbb Z$. More precisely, the canonical map $\mathbb Z/2020\mathbb Z\to T^{-1} (\mathbb Z/2020\mathbb Z)$ is surjective and all you need is to find its kernel.