# What letter should I use to denote an ideal?

In commutative algebra, there seem to be two rather different notational conventions for ideals: either $I,J, \dots$ or $\mathfrak{a}, \mathfrak{b}, \dots$.

By itself, it is hardly surprising - after all, lots of other notations vary from source to source. I have, however, come across both conventions in a single script or article on many occasions. And the difference is rather striking: different letter is used, different case, different font...

I find this rather surprising and a little confusing, since usually a single convention is used within a piece of writing. I would very much appreciate any information on where such notational complication comes from. Is one of these conventions preferable (in some circumstances)?

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I learned with $I,J$... – Belgi Sep 21 '12 at 18:09
I like $I$ and $J$ because they require less characters to typeset than $\mathfrak{a}$...although I like $\mathfrak{p}$ for prime ideals over, e.g., $P$. – Keenan Kidwell Sep 21 '12 at 18:41
Ideally, $\mathfrak{i}$, – Hans Engler Sep 21 '12 at 21:49

Since Bourbaki in His Commutative Algebra and His prophets Dieudonné and Grothendieck in EGA use Fraktur, you have no choice but to do the same.
I shudder to think of the fate befalling heretics and miscreants who would incur His ire by using Latin letters.

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A very poetic answer, and yet quite informative. I'd say following conventions set by Dieudonné and Grothendieck sounds like a good idea. – Jakub Konieczny Sep 22 '12 at 8:28
Thanks, Feanor: I'm glad you agree with the answer. – Georges Elencwajg Sep 22 '12 at 10:46

If you are clear when you define things, and use the same convention consistently, and you don't go out of your way to make the notation ridiculous, nobody cares.

I think whatever you pick (within reason) you'll do fine.

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Well, I care ;) I might be in a minority here, but I do believe a well chosen notation makes understanding a piece of mathematics easier. – Jakub Konieczny Sep 21 '12 at 19:18
@Feanor If you ask me, just never use $\mathfrak{A}$. I think it looks terrible. – rschwieb Sep 21 '12 at 20:14
I won't, I also dislike it. In particular, it is not even obvious which letter it represents unless you know it in advance (it could well be U as well as A, or even M). – Jakub Konieczny Sep 22 '12 at 8:23