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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.

Community
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ymar
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  • 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. – rschwieb May 21 '12 at 23:07
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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 00:58
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    The ideal usage is much rarer than groups in my experience. As such, it may confuse many readers. – Bill Dubuque May 22 '12 at 02:59
  • 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...) – Asaf Karagila May 22 '12 at 07:17
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    My argument for $\lhd$ over $\triangleleft$ is that it's four characters instead of 13. – Adelaide Dokras May 22 '12 at 07:39
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    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 08:07
  • @WillieWong I think this is a meta question. I'm asking specifically about what notation I should use on M.SE. –  May 22 '12 at 09:02
  • @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?! – user1729 May 22 '12 at 09:11
  • 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... – Willie Wong May 22 '12 at 11:09
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    @Willie, is it a peninsula? – Gerry Myerson May 22 '12 at 12:34
  • @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. – Bill Dubuque May 22 '12 at 13:29

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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.

Martin Sleziak
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Nobody
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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 05:22
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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.)

Jonas Meyer
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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 $\mathscr R$, $\mathscr L$,... – Nobody May 22 '12 at 09:44
  • @scaaahu I have never seen this notation used in semigroup theory either. I was asking about ring-theoretic ideals. –  May 22 '12 at 12:44
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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.

Eugene
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