# What does w.l.g mean?

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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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

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 $p|b$.