9
$\begingroup$

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.

  • 0
    I think it's standard for ring theory. In any case, we have to deal with much worse notational overlap as a matter of routine.2012-05-21
  • 3
    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.2012-05-22
  • 4
    The ideal usage is *much* rarer than groups in my experience. As such, it may confuse many readers.2012-05-22
  • 0
    I saw this more than a few times around here, but I also saw people that did not use it and wrote "is an ideal of ..." explicitly. As for typesetting, I believe that $\lhd$ has a better spacing as a relation but this is just a belief. (Let us put it to the test: $$I\lhd R\quad I\triangleleft R$$ seems the same to me, I guess I was wrong...)2012-05-22
  • 4
    My argument for $\lhd$ over $\triangleleft$ is that it's four characters instead of 13.2012-05-22
  • 6
    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](http://tex.stackexchange.com/q/38982/9954). \\ Although I am not entirely sure that everything (spacing in particular) works in mathjax exactly the same way as in TeX.2012-05-22
  • 0
    @WillieWong I think this is a meta question. I'm asking specifically about what notation I should use on M.SE.2012-05-22
  • 0
    @BillDubuque: do you mean that people just haven't come across the notion of an ideal yet? Or do you mean there is another way of writing "is an ideal of" that noone has told me about?!2012-05-22
  • 0
    Conventions on MSE generally follow that of the general mathematics community. If the symbol is generally accepted as a common notation in the literature, I don't see why it won't be on MSE. Conversely, if the symbol is generally not accepted, I don't see why you can assume it to be a common notation on MSE. In other words, MSE is not an island...2012-05-22
  • 4
    @Willie, is it a peninsula?2012-05-22
  • 0
    @user1729 I mean that said notation is used far less commonly in ring theory literature than in group theory literature. In fact, so infrequently that I cannot recall the last time I saw it used.2012-05-22

3 Answers 3

3

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.

  • 0
    And according to wiki there seems to be no difference between the two symbols listed.2012-05-23
12

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.

  • 8
    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.2012-05-23
10

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.)

  • 0
    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 $\mathscr R$, $\mathscr L$,...2012-05-22
  • 0
    @scaaahu I have never seen this notation used in semigroup theory either. I was asking about ring-theoretic ideals.2012-05-22