# What does $\langle x,3\rangle$ mean if I'm talking about $\mathbb Z[x]$

I know this sounds like a basic question but I'm really confused. What does the notation $\langle x,3\rangle$ refer to for $\mathbb Z[x]$? Can someone write out what this is?

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The book you're using ought to have defined the symbol somewhere... where did you see it? – J. M. May 13 '11 at 8:35
This is on my practice exam for Algebra. We don't have a book. We rely on the professor's notes. – Person May 13 '11 at 8:35
If you see unfamiliar notation on a practice exam given to you by your instructor, why wouldn't you ask your instructor about it? If indeed this notation was not covered in class, then other students are going to have the same question. If it was covered in class, then you missed it, which is potentially useful information: did you miss a day or more of class? Are you taking incomplete notes? Are there other things you missed as well? Notice that you can't answer these questions by posting to an internet Q&A site... – Pete L. Clark May 13 '11 at 10:13
I just noticed that the OP asked a question last week in which the ideal $\langle 2,x \rangle$ appeared, with definition. So...yes, s/he does seem rather confused. Again I will recommend contacting the instructor, although looking at the same previous question it seems the OP may not want to do that, which is an unfortunate situation. – Pete L. Clark May 13 '11 at 10:41
It's a mystery. – quanta May 13 '11 at 16:27

The notation means the ideal of $\mathbb{Z}[x]$ generated by the elements $3$ and $x$ (see here) $$\langle x,3\rangle=\{xf+3g\mid f,g\in\mathbb{Z}[x]\}=\{a_nx^n+\cdots+a_1x+a_0\mid a_i\in\mathbb{Z}\text{ and }3\text{ divides }a_0\}.$$
$\rm \langle x,3\rangle\:$ could mean a few things. Here are some possibilities, listed most-likely first. It could denote the ideal $\rm\: x\ \mathbb Z[x] + 3\ \mathbb Z[x]\:.\:$ Or it could denote the additive subgroup $\rm\:x\ \mathbb Z + 3\ \mathbb Z\:.\:$ Finally it could denote $\rm\:gcd(x,3)\:,\:$ though the gcd is more commonly denoted as $\rm\:(x,3)\:$ or $\rm\:[x,3]\:.$