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In Herstein’s Topics in Algebra, p.147, there is a sentence

If $\pi$ in $R$ is a prime element of $R$ and $a \in R$, then either $\pi \mid a$ or $(\pi,a) = 1$, for, in particular, $(\pi,a)$ is a divisor of $\pi$ so it must be $\pi$ or $1$ (or any unit).

I’m having a bit of difficulty seeing why the last part of this sentence is true (“for, in particular…”).

Note:

  • Herstein’s definition of a prime element in a Euclidean ring $R$ is a nonunit $\pi$ such that whenever $\pi = ab$, where $a, b$ are in $R$, then one of $a$ or $b$ is a unit in $R$.
  • This result comes before the Unique Factorization Theorem, so that can’t be used. However, we have proved that every element of a Euclidean ring $R$ is either a unit or can be written as a product of finitely many primes. (I don’t know if this is useful, but it’s what I tried to use, to no avail.)

My attempt:

I tried supposing to the contrary, so that $(\pi,a)$ is neither $\pi$ nor $1$ nor a unit. Since $(\pi,a)$ divides $\pi$ we can write $\pi = (\pi,a)k$ for some $k \in R$. Then by definition of $\pi$ being prime, $k$ must be a unit, since $(\pi,a)$ isn’t. So $\pi$ and $(\pi,a)$ are associates. This is where I tried a few other things and got stuck each time.

twosigma
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1 Answers1

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Hint: a prime $\pi$ is irreducible so its divisor $\,(\pi,a)\,$ is either a unit, or else associate to $\pi$ (then $\,\pi\mid (\pi,a)\Rightarrow\, \pi\mid a)$

Bill Dubuque
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  • Recall: $ $ an irreducible $,\pi,$ is a nonzero element having only trivial factors, i.e. $\ d\mid \pi,\Rightarrow, d\sim 1,$ or $,d\sim \pi,,$ i.e. $,d,$ is either a unit or a unit times $\pi,,$ i.e. a "trivial" factor of $,\pi\ \ $ – Bill Dubuque Jan 15 '21 at 20:42
  • Ok, so when Herstein says "...$(\pi,a)$ is a divisor of $\pi$ so it must be $\pi$ or $1$ (or any unit)", for the case where it is $\pi$, he does not literally mean that $(\pi,a)$ has to be the element $\pi$, but only that it is associate to $\pi$? That was my main confusion. – twosigma Jan 15 '21 at 21:08
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    @twosigma Yes, that's correct. – Bill Dubuque Jan 15 '21 at 22:36