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Ireland and Rosen state in A Classical Introduction to Modern Number Theory: "We now know that in $\mathbb{Z}$ and $k[x]$ there are infinitely many nonassociate primes. It is instructive to consider a ring where all primes are associate, so that in essence there is only one prime."

They go on to say that if $p\in\mathbb{Z}$ is a prime, then $\mathbb{Z}_p$ is the set of all rational numbers $a/b$ so that $p\nmid b$.

What does the statement "... in essence there is only one prime" mean? Why is this kind of ring helpful?

user26857
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    It means there is only one prime up to associates, i.e. you've killed all primes $\neq p$ by adjoing their inverse, so only $p$ remains (and all its associates $pn\neq 0$ for $,p\nmid n).,$ This is a special case of localization. – Bill Dubuque Jun 18 '21 at 00:51
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    One reason such localizations are useful here is that for many divisibility problems we can reduce "global" problems to simpler "local" problems in these localizations - analogous to how we solve many problems in $\Bbb Z$ by working one prime at a time. – Bill Dubuque Jun 18 '21 at 01:53
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    A general example is the class of Krull domains, which include UFDs, Dedekind domains, and Noetherian integrally closed domains. Here the localizations $D_{\cal P}$ at minimal prime ideals $\cal P$ are DVRs = discrete valuation rings where divisibility is particular simple because ideals are totally ordered by inclusion, as in your example $,up^j\mid vp^k\iff (p^j)\supset (p^k) \iff j\le k.\ $ – Bill Dubuque Jun 18 '21 at 01:53

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Two elements $a,b$ of a ring $R$ are said to be associate if $a\mid b$ and $b\mid a$. So, for example, in $\mathbb{Z}$, $7$ and $-7$ are associate. In this case (obviously) we can always write $b = ua$ for some unit $u$.

Now, on to your question. If all primes in a ring $R$ are associate, we can say that there is really a unique prime element $p$, and the rest are some unit multiple of this prime element (can be written as $up$ for some unit $u$). So, in this sense, we can say that indeed there is only one prime, because note that there is only one prime ideal (the one generated by $p$).

Why is it useful to study these kinds of rings? Well, it depends, obviously. I can’t give you a satisfactory answer. However, note that some really unique things happen in these kinds of rings. For example, if the Fundamental Theorem of Arithmetic holds (i.e., we are in a UFD) then every element of the UFD can be expressed as a power of $p$ times a unit.

I hope my answer helped.

Samuel M. A. Luque
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    It did help, thank you so much. –  Jun 18 '21 at 01:12
  • Would this be a working example? Consider the ring $\mathbb{Z}_7$. $4/29$ is a unit of $\mathbb{Z}_7$ since $4/29\cdot29/4=1$. Does the "power of $p$ times a unit" apply here in the sense that$4/29=7^0\cdot4/29$? –  Jun 18 '21 at 01:27
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    @DullandWitlessBoi Yes, that's correct. – Ravi Fernando Jun 18 '21 at 02:28
  • Thanks! It's been a while since I thought in terms of abstract algebra. –  Jun 18 '21 at 03:58