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My lecturer has abbreviated something and I can't find what it is. Here is how it has come up in my notes:

Lemma: In any integral domain, $p$ prime $\implies p$ irreducible.

Proof: Suppose $p$ is prime and $p = ab$. w.l.g $p \mid b$. So $cp = b$ for some $c \in R$...

What does that w.l.g bit mean? Does it make sense from there or shall I try and find another example?

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4  
Why didn't you asked your lecturer while taking notes? –  Mr.ØØ7 Apr 10 '13 at 12:07
    
It didn't come up in the lecture. I thought I made some mistake in the proof from what she said so I went through her online notes and I saw it there –  Kaish Apr 10 '13 at 12:09

2 Answers 2

up vote 9 down vote accepted

W.l.g is a reasonably common abbreviation for "without loss of generality" (used for assuming something without affecting the validity of the proof in general, usually because there are multiple, identical alternatives). Another common one is WLOG.

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WLOG was in vogue when I was in grad school. –  ncmathsadist Apr 10 '13 at 12:50
    
It used to mean Weiss logarithm :-) –  Julián Aguirre Apr 10 '13 at 12:58
3  
I have never seen Wlg, but only WLOG. –  KCd Apr 10 '13 at 13:07

It means "without loss of generality". In this case we have $p|ab$, so either $p|a$ or $p|b$. In the former case we can just switch the names of $a$ and $b$ so that $p|b$ again, so there is no need to consider the case where $p|a$ ; and we can assume "without loss of generality" that $p|b$.

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