Are the authors sloppy when forgetting the property “with unity” of $\mathbb Z$ in Remark (b)

proof-explanationrational numbersring-theory

I am reading Section 9. The Rational Numbers from textbook Analysis I by Amann/Escher. They present two theorems:

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and then say

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It is stated in Theorem 9.1 that $\mathbb Z$ is a smallest domain with unity, but it is stated in Remark (b) that "all that was necessary was that $\mathbb Z$ be a domain" and that "any domain $R$ is a subring of
a unique (up to isomorphism) minimal field $Q$".

My question: Are the authors sloppy when forgetting the property with unity of $\mathbb Z$ and $R$ in Remark (b)?

Best Answer

Multiplicative identity is not necessary for the construction of field of fractions. If $R$ is integral domain, and $Q$ its field of fractions, $R$ embeds into $Q$ by $r\mapsto \frac{rs}s$, for any $s\in R\setminus\{0\}$ (and doesn't depend on choice of $s$).

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