# Is $\lhd$ common notation for “is an ideal of”?

This question is because of this comment. I would like to know if I should refrain from using $\lhd$ for "is an ideal of" in ring-theoretic questions. Is it common enough, or should I explain what it means? It's standard notation where I study, but I understand it doesn't have to be everywhere.

Also, is there any strong argument for choosing either $\lhd$ or $\triangleleft$? I don't know much about math typesetting, and I've noticed that people can give surprising and convincing arguments for using one symbol and not using another.

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It's common for "is a normal subgroup of", and I guess "ideal" is to rings as "normal" is to groups, so I can see why people would use it. But it's new to me. – Gerry Myerson May 22 '12 at 0:58
The ideal usage is much rarer than groups in my experience. As such, it may confuse many readers. – Bill Dubuque May 22 '12 at 2:59
My argument for $\lhd$ over $\triangleleft$ is that it's four characters instead of 13. – Adelaide Dokras May 22 '12 at 7:39
It seems that \lhd is more suitable for relations and \triangleleft for binary operations. So in this usage \lhd seems to be better.\\ Namely I found in amstex.sty: \DeclareMathSymbol\vartriangleleft {\mathrel}{AMSa}{"43} \global\let\lhd\vartriangleleft and in fontmath.ltx: \DeclareMathSymbol{\triangleleft}{\mathbin}{letters}{"2F}. Difference between mathrel and mathbin is explained here. \\ Although I am not entirely sure that everything (spacing in particular) works in mathjax exactly the same way as in TeX. – Martin Sleziak May 22 '12 at 8:07
@Willie, is it a peninsula? – Gerry Myerson May 22 '12 at 12:34

It would be a good idea to define the notation if you use it. It is not standard notation everywhere.

(I don't know which typesetting is preferred.)

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Echo to @Jonas Meyer. I checked serveral semigroup books I have. I don't see anyone uses those notations. Maybe I missed it. If I did, it means the notations are not used often. Otherwise I would have seen it. So, it's not standard notations in semigroup theory as opposed to $\mathscr R$, $\mathscr L$,... – scaaahu May 22 '12 at 9:44
@scaaahu I have never seen this notation used in semigroup theory either. I was asking about ring-theoretic ideals. – user23211 May 22 '12 at 12:44

I swore I saw this notation somewhere and it took me over an hour to find it.

It's in P.M. Cohn's Introduction to ring theory 2000 ed.

On page 12, he says "One often writes $A \lhd R$ to indicate that $A$ is an ideal of $R$.".

I cannot tell which symbol he used for typesetting.

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Heh-heh, saying something is often done is one way to convince a reader to do it. That notation is pretty rare in my experience. – KCd May 23 '12 at 5:22

I have checked several textbooks (such as Reid's commutative algebra, Dummit's abstract algebra and even Isaac's Algebra) and none of them seem to use the notation you are suggesting. So it doesn't look standard to me.

EDIT: As usual Wiki comes to the rescue! In this wiki page is says that the symbol is used to denote normal subgroups, IDEALS, and the antijoin. So it must be somewhat "standard". Maybe it was popular in the past but not anymore.

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And according to wiki there seems to be no difference between the two symbols listed. – Eugene May 23 '12 at 3:00