I am doing exercise problem of chapter 7 of Dummit and Foote's Algebra.
The problem states
7.5.2 Let $R$ be an integral domain and let $D$ be a nonempty subset of $R$ that is closed under multiplication. Prove that the ring of fractions $D^{-1}R$ is isomorphic to a subring of the quotient field of $R$. (hence is also an integral domain.).
In this problem, there is no restriction for $D$. For example, $D$ can contain $0$. But, when Dummit and Foote were constructing ring of fraction (Theorem 15), they specifically said $D$ should not contain $0$. Is it just wrong statement or I am missing something?
Best Answer
$ D $ obviously cannot have zero because these are the denominators of the quotient field you are constructing! They are using the notation of Theorem 15, I think.