9
$\begingroup$

I'm trying to understand a proof of Hungerford's book which says that in a integral domain every prime element is irreducible:

I didn't understand why this implication $p=ab\implies p|a$ or $p|b$, is not the contrary $p=ab\implies a|p$ and $b|p$ ?

I'm a little confused

Thanks in advance

$\endgroup$

3 Answers 3

Reset to default
12
$\begingroup$

If $p = ab$, then in particular $p$ divides $ab$, because $ab = p \cdot 1$. Since $p$ is prime, it has to divide either $a$ or $b$.

$\endgroup$
7
  • $\begingroup$ Of course,thank you very much for your help $\endgroup$
    – user74141
    May 29, 2013 at 14:26
  • $\begingroup$ @user74141, you're welcome. $\endgroup$
    – Andreas Caranti
    May 29, 2013 at 14:29
  • 1
    $\begingroup$ @user74141 One can make the inference prime $\,\Rightarrow\,$ irreducible more clear by using an alternative definition of irreducible - see my answer. $\endgroup$
    – Key Ideas
    May 29, 2013 at 16:40
  • $\begingroup$ I don't understand the part, 1=xb hence b=1 (or b is a unit), how does it tie together, can you help? $\endgroup$
    – ciceksiz kakarot
    Nov 18, 2013 at 19:10
  • 1
    $\begingroup$ @utkarshk5, I subscribe to the persuasion that integral domains are defined to have a unity. I know some authors prefer otherwise - should have mentioned my preference explicitly. $\endgroup$
    – Andreas Caranti
    Jan 11, 2021 at 15:34
8
$\begingroup$

Prime $\Rightarrow$ irreducible is obvious if we employ a definition of irreducible in associate (vs. $\rm\color{#0a0}{unit}$) form. Then $\,\color{#c00}{ p=ab\,\Rightarrow\, p\mid ab}\,$ immediately yields the sought inference, as follows.

Theorem $\ \ $ In the following, $\,\ (1)\,\Rightarrow\,(2)\!\iff\! (3)$

$(1)\ \ \ \color{#c00}{p\ \mid\ ab}\ \Rightarrow\ p\:|\:a\ \ {\rm or}\ \ p\:|\:b\quad$ [Definition of $\:p\:$ is prime]

$(2)\ \ \ \color{#c00}{p=ab}\ \Rightarrow\ p\:|\:a\ \ {\rm or}\ \ p\:|\:b\quad$ [Definition of $\:p\:$ is irreducible, in associate form]

$(3)\ \ \ p=ab\ \Rightarrow\ a\:|\:1\ \ {\rm or}\ \ b\:|\:1\quad$ [Definition of $\:p\:$ is irreducible, in $\rm\color{#0a0}{unit}$ form]

Proof $\ \ \ (1\Rightarrow 2)\,\ \ \ \color{#c00}{p = ab\, \Rightarrow\, p\mid ab}\,\stackrel{(1)}\Rightarrow\,p\mid a\:$ or $\:p\mid b.\ $ Hence prime $\Rightarrow$ irreducible.

$(2\!\!\iff\!\! 3)\ \ \ $ If $\:p = ab\:$ then $\:\dfrac{1}b = \dfrac{a}p\:$ so $\:p\:|\:a\iff b\:|\:1.\:$ Similarly $\:p\:|\:b\iff a\:|\:1.$

Beware that factorization theory is more complicated in non-domains. Basic notions such as associate and irreducible bifurcate into a few inequivalent notions. See here for more on that.

$\endgroup$
3
$\begingroup$

I think the OP was asking why the first implication is there instead of the alternate implication he proposed. The answer is that the first implication is part of the definition of a prime. Look at the definition of prime that precedes the theorem in question in Hungerford: An element p of R is prime provided that: (i) p is a nonzero non unit; (ii) p|ab $\Rightarrow$ p|a or p|b.

$\endgroup$
1
  • $\begingroup$ The link for the book is not working, can i ask exactly what book this was and the page number of the proof? $\endgroup$
    – user531499
    Feb 17, 2019 at 14:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.