Assume $S$ is a commutative integral domain, and $R \subseteq S$ is a subring. Assume $S$ is finitely generated as an $R$-module, i.e., there exist elements $s_1, \ldots, s_n \in S$ such that $S = s_1R + s_2R + \cdots + s_nR$.

(a) Show that $R$ is a field if and only if $S$ is a field.

(b) Is the statement true if the assumption that $S$ is an integral domain is dropped?

I haven't been able to do either direction.

Starting with the forward direction I assume $R$ is a field and consider a non-zero element $s ∈ S$ and write $s = s_1r_1 + · · · + s_nr_n$ then subtract the sum from both sides to get $s - s_1r_1 + · · · + s_nr_n = 0$. I want to try to use the integral domain fact by somehow writing the left side as a product like so: $s(1 - s^{-1}(s_1r_1 + · · · + s_nr_n)) = 0$ and since s is non-zero $1 - s^{-1}(s_1r_1 + · · · + s_nr_n) = 0$ and therefore $s^{-1}s = 1$ but there are obvious problems with that argument. Any help or hints would be appreciated. This is not a homework problem I am studying for a qual.


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