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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)?

Akira
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  • It is currently a common though not universal practice to treat ring as having a multiplicative identity: see https://math.stackexchange.com/questions/48587/definition-of-ring-vs-rng/48612 (some people call the structure without a multiplicative identity a rng) and so also a domain. Perhaps they mentioned this in 9.1 to be helpful but did not in 9.2 as they felt they had already done so. – Henry Jun 08 '19 at 08:22
  • @Henry ... and as the question shows, this attempt to being helpful can turn out to be confusing instead. -- For clarity, we woul dneed to see the sections of the book where the corresponding definitions are made – Hagen von Eitzen Jun 08 '19 at 08:24
  • @HagenvonEitzen which corresponding definitions do you like to see in my textbook? I will update my post then. – Akira Jun 08 '19 at 08:29

1 Answers1

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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 $s$).

Ennar
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    We should note that "integral domain" commonly means non-trivial commutative ring without zero-divisors, and the non-triviality condition was (sloppily) left out in Theorem 9.1 – Hagen von Eitzen Jun 08 '19 at 08:29