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In $\textit{A Classical Introduction to Modern Number Theory}$, the authors define $\mathbb{Z}_p$ as a set where $p$ is a prime number and $a,b$ form the rational number $a/b$ such that $p\nmid b$.

After describing this set, the authors mention "One easily checks using the remark following Corollary 1 to Proposition 1.1.1 that $\mathbb{Z}_p$ is a ring.

That remark is "If $p$ is a prime and $p\nmid b$ and $p\nmid c$, then $p\nmid bc$.

I'm puzzled by the semantics of suggesting the use of the remark to the corollary. I understand how this applies: if I have elements $a/b,c/d$, where $p\nmid b$ and $p\nmid d$, then $a/b+c/d=(ad+bc)/bd\Rightarrow p\nmid bd$. A similar calculation is done with multiplication. This definitely shows $\mathbb{Z}_p$ is closed.

For the sake of context, the next paragraph demonstrates $a/b$ is a unit implies $p\nmid a$ (and the converse) and all elements fo $\mathbb{Z}_p$ can be expressed as a power of $p$ times a unit. The conclusion is that all primes of $\mathbb{Z}_p$ are associate. (I'm totally fine with this part.)

Does the phrasing indicate that I use this corollary remark to justify the work showing why the other ring properties are true, or is it an end-all statement saying "because this is satisfied, it's a ring"?

Bill Dubuque
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  • The ring is $\Bbb Z_{(p)}$, and not the ring of $p$-adic integers $\Bbb Z_p$. As a localisation of the ring $\Bbb Z$ is it a ring, see wikipedia. – Dietrich Burde Jun 21 '21 at 18:15
  • The remark is used in an (implicit?) application of the subring test - see the linked dupe. – Bill Dubuque Jun 21 '21 at 19:10
  • $\mathbb{Z}p$ is pretty common notation for the ring of integers mod $p$. Maybe it shouldn't be, but it definitely is used frequently. Probably $\mathbb{Z}/p\mathbb{Z}$ is the clearest notation, since $\mathbb{Z}{(p)}$ sure looks a lot like $\mathbb{Z}_p$. – Eric Snyder Jun 21 '21 at 19:13

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$\mathbb{Z}_p$ is a subset of the rational numbers, so all of the ring axioms are satisfied as long as the sum and product of elements of $\mathbb{Z}_p$ are contained in $\mathbb{Z}_p$, and 1 is contained in $\mathbb{Z}_p$. The author means that the aspects of showing that $\mathbb{Z}_p$ is a ring (once you know this theorem) are implied by the remark.

Quizzical
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  • Yes! Thank you so much. That makes perfect sense, and consequently saves 5-10 minutes of my presentation. It's wonderful how "lazy" math is: always looking for a stronger, quicker way to justify something. –  Jun 21 '21 at 16:51
  • Please strive not to add more dupe answers to dupes of FAQs, cf. recent site policy announcement here. – Bill Dubuque Jun 21 '21 at 19:11
  • Where do I check if something is a duplicate? I use the general search box from the MSE homepage and then check after I begin typing a question. Oftentimes I don't see anything similar pop up there. –  Jun 21 '21 at 19:30
  • Searching on "fraction subring test" does the trick. Note that you need to write "username" for a user to be notified by your comment, where is the atsign character (I discovered your comment only by luck) – Bill Dubuque Jun 21 '21 at 19:51