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Is this question acceptable?: Why do algebra texts almost always DEFINE rings to be associative?

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    • CommentAuthorAndrewL
    • CommentTimeOct 25th 2010 edited
     

    I may have asked this question already-memory fails me. So just in case, I'm running it past meta first. It's a question that I've always asked myself in basic algebra and I'm asking it again now that I'm getting ready for my Master's orals in the subject. Despite nonassociative rings and algebras (i.e. octonions,Lie rings/algebras and Jordan rings/algebras) being major classes of rings,they are almost always defined as necessarily associative in the multiplication operation. To quote Lang's ALGEBRA,3rd edition,chapter II:

    Definition: A ring A is a set, together with two laws of composition called
    multiplication and addition respectively, and written as a product and as a sum
    respectively, satisfying the following conditions: R 1. With respect to addition, A is a commutative group. R 2. The multiplication is associative, and has a unit element. R 3. For all x,y,z e A we have (x + y)z = xz + yz and z(x + y) = zx + zy. (This is called distributivity.)

    Thus in Lang-and most other texts-rings are required to be associative in the second operation and in algebras,an analogous axiom to R2 holds for the vector space structure in most texts. There are a FEW exceptions-the major texts that come to mind are Nathan Jacobson's BASIC ALGEBRA and I.R.Herstien's TOPICS IN ALGEBRA (from which I first learned the subject several years ago). Recently,we can add Louis Rowen's 2 volume treatise to that list. Not surprisingly,these texts are all by prominent ring theorists-Rowen,of course,was a doctorial student of Jacobson's at Yale.I'm not sure,but I believe Claude Chavalley's FUNDAMENTAL CONCEPTS OF ALGEBRA also defines rings nonassociatively. Those exceptions aside-if you choose a dozen of the legion of currently available algebra books,chances are all of them will define rings the way Lang does. But since nonassociative rings form such a major class of rings-particularly in Lie theory with the rise of noncommutative geometry-why aren't general rings defined in modern textbooks omitting the associativity requirement and simply making the statement that most rings encountered in "real life" are associative? Algebraicists want to chime in on this?

  2.  
    You asked it here :
    http://mathoverflow.net/questions/32340/what-is-the-correct-definition-of-a-ring-closed
    It was closed. My opinion of it has not changed.
  3.  
    It's just a convention. Textbooks define manifolds to not have boundary. And almost immediately after that they define "manifolds with boundary". In most textbooks "manifolds with boundary" technically aren't manifolds, because manifolds do not have boundary. Similarly "manifolds with corners" generally are not "manifolds with boundary". And "stratified spaces" are generally not "manifolds with corners", etc.

    Sometimes people prefer to keep terminology specific to the original motivation. As ideas generalize, you tack on different modifiers to the descriptive word and consider them new objects. Similarly, the Dirac delta function isn't really a function (at least, not in the standard way). And regular values of functions do not need to be in the image of functions -- i.e. they're not actually values of the function, etc...
    • CommentAuthorAndrewL
    • CommentTimeOct 25th 2010
     

    Yeah,that's a memory I wanted to relive.

    And my opinion hasn't changed either,Andy-I think you're wrong.

    But I was overruled,so that's that. Next.

    Andrew L.

  5.  
    @Andy, that's weird.

    Andrew, do I understand you correctly in that you want to ask your closed question again? IMO it would have been better to start this thread with a link to your closed, essentially identical question.
  6.  
    Another point here is that most algebra texts do not discuss any interesting examples of nonassociative algebra (eg none that I know of spend any time on Lie algebras). It would thus be tiresome to constantly put the word "associative" there.

    For another example of this, there exist people who study manifolds that aren't Hausdorff (for instance, they show up quite a bit when studying foliations). However, the majority of people who work with manifolds only study Hausdorff ones, so the convention is to include that in the definition. If you care about non-Hausdorff manifolds, you have to specify that in the beginning of your papers.
  7.  
    @Ryan : What's weird?
  8.  
    @Andy: I thought it rather weird to request to start a new thread with the same topic as a previously-closed thread, and to avoid saying that's what you're trying to do.
  9.  
    @Ryan : My reading of the situation is that Mr. L couldn't remember if he had asked the question before (in fact, he states this in his post) and is not computer-literate enough to figure out how to browse the questions he had already posted.
  10.  

    Lang's convention in this case seems completely reasonable: An algebra is an abelian group equipped with a certain bilinear map, and a ring is an associative algebra with unit. We have two distinct words, so we might as well use them to indicate distinct things, and in doing so, we maintain some consistency with established research literature. I don't think many people who study not-necessarily-associative algebras are particularly troubled by the fact that the term "ring" is not typically applied to describe the objects they study. According to Google, "algebra" wins over "ring" by a large margin when either "Lie" or "Jordan" are added, so I suspect there isn't much of a fight here.

    • CommentAuthorMariano
    • CommentTimeOct 25th 2010
     

    I've seen "Lie ring" to be used in opposition to "Lie algebra", but I don't recall what the difference was...

    • CommentAuthorEmerton
    • CommentTimeOct 25th 2010
     

    Dear Mariano,

    My memory is that "Lie ring" means "Lie algebra over a (commutative, I guess) ground ring other than a field". Does that seem reasonable?

    Best wishes,

    Matt

    • CommentAuthorAndy Putman
    • CommentTimeOct 25th 2010 edited
     
    @Mariano, Emerton : I frequently have to deal with the Lie algebra structure obtained from the lower central series of a group (which has ground ring Z), and this is definitely sometimes referred to as a Lie ring in the literature. I can't attest to other uses of the term.
    • CommentAuthorMariano
    • CommentTimeOct 25th 2010
     

    The context was most probably that of the lower central series of a group, so that's it.

  15.  

    Now I am rather curious about the history of the term "Lie ring". My unsubstantiated guess is that the first person to use it subscribed to the convention (which I have seen reasonably often) that algebras must be defined over a field.

    • CommentAuthorBCnrd
    • CommentTimeOct 26th 2010
     
    I've never heard of the phrase "Lie ring" before in my life, nor the convention that algebras are defined over a field. (The case of "central simple algebra" is a red herring, since the "simple" aspect immediately restricts the possibilities of a commutative ground ring very severely.) One has Azumaya algebras over rings, and Lie algebras over rings or sheaves of rings (pervading SGA3, for example). Since the context should always make the base ring clear, what is lost by always saying "Lie algebra"?
    • CommentAuthorAndy Putman
    • CommentTimeOct 26th 2010 edited
     
    @BCnrd : I think the term is defensible in certain cases. It would be silly to call a general commutative ring an algebra over Z, wouldn't it? In the same way, there are examples of Lie-algebra type structures where there isn't any natural field/ring floating around to serve as scalars (other than the trivial Z). For eg, the Lie ring associated to the lower central series of a group.
    • CommentAuthorEmerton
    • CommentTimeOct 26th 2010
     

    Dear Brian,

    I think that Andy is right, and that this terminology comes up in (perhaps older?) texts on nilpotent groups and such, to mean a Lie algebra over the integers. Again, although I can't cite a text, it wouldn't surprise me if at some point algebra was always taken to mean "algebra over a field". (For example, what terminology did Artin, Noether, or van der Waerden use?)

    Best wishes,

    Matt

  19.  

    I've seen "algebra" defined with the assumption of a field of scalars in linear algebra books, for example Hoffman and Kunze. H&K also defines "ring:, so the more restrictive definition of algebra is not just for the sake of avoiding that.

    • CommentAuthorBCnrd
    • CommentTimeOct 26th 2010
     
    Dear Andy: I don't see anything silly about the concept of a Z-algebra. For example, "finite type Z-scheme" is a phrase that comes up a lot in arithmetic geometry, in the course of which one often says things like "finitely generated Z-algebra" (rather than "finitely generated ring"). Likewise, if I were handed a Chevalley group G (any of those famous smooth affine Z-group schemes) I wouldn't hesitate to refer to Lie(G) as a "Lie algebra over Z" (and SGA3 does likewise, as do many other references of that flavor). I trust you and Matt know the earlier literature and traditions better than me. Certainly in linear algebra texts it is reasonable that everything is over a field, given the background of readers of such books. My only point was that there are some parts of algebraic geometry in which "Lie ring" seems to never be used and "Lie algebra" is used, and that I didn't see a compelling reason why nowadays one would have a quibble about saying "Lie algebra" over any coefficient ring (perhaps in contrast with generations ago). For instance, look at the very bottom of page 1 of Humphreys' book on Lie algebras, or sections 1.1 and 1.2 of Chapter I of Bourbaki LIE.
    • CommentAuthorEmerton
    • CommentTimeOct 26th 2010
     

    Dear Brian,

    I agree that Lie ring sounds, and probably is, old-fashioned. I also agree with you regarding Z-algebras (just as I frequently say Z-module rather than abelian group). I wonder though if this is a symptom of being an arithmetic geometer/number theorist?

    Best wishes,

    Matt

  22.  
    @BCnrd : I also agree that the phrase "Lie ring" is a bit old-fashioned, and I personally would not use it (the papers I am thinking of that use it are quite old and by eg Magnus and Philip Hall). As far as Z-algebras vs commutative rings, I would only use the phrase "Z-algebra" if "Z" were a variable or if I really wanted to emphasize the Z-action. But as Emerton said, this is probably a function of the area of math I work in.
    • CommentAuthorKevin Lin
    • CommentTimeOct 26th 2010 edited
     

    But since nonassociative rings form such a major class of rings-particularly in Lie theory with the rise of noncommutative geometry-why aren't general rings defined in modern textbooks omitting the associativity requirement and simply making the statement that most rings encountered in "real life" are associative?

    @AndrewL: To me -- as someone more geometrically inclined, and also as a nonexpert -- it feels somewhat "wrong" to study associative rings (or associative algebras) and Lie algebras as objects living in the same category of "nonassociative rings" (or "nonassociative algebras"). Commutative rings have a good corresponding geometric theory (i.e. scheme theory); associative rings perhaps retain some of that geometry (see the MO question I asked on "noncommutative algebraic geometry"); I don't know this for a fact, but I'd guess that probably very little if any of that geometry survives the passage to "nonassociative rings".

    Lie algebras do of course have connections with geometry, but these connections do not seem to be very similar or analogous to the connections between commutative rings and geometry.

    Moreover, have you ever seen -- for instance -- a meaningful or interesting morphism from an associative algebra to a Lie algebra, or from a Lie algebra to an associative algebra? I haven't...

    • CommentAuthorKevin Lin
    • CommentTimeOct 26th 2010 edited
     

    @Andy Putman: For example, given a morphism of commutative rings A -> B, it seems natural to say that it makes B into an A-algebra, no?

    Since in arithmetic geometry one is interested in -- say -- schemes over Spec Z, it seems therefore natural that terms like "Z-algebra" would be common, if for nothing else but to emphasize that one is working over Spec Z...

  25.  
    @Kevin : Yes, I certainly think it is reasonable in arithmetic geometry to say things like "Z-algebra", both because "Z" is a variable (you might want to make a base change) and because you want to emphasize the Z-action (ie you are working over Z).

    However, in other contexts it would be silly. For instance, I do quite a bit of group theory, and I think I would be laughed at if I called the abelianization of a group a "Z-module" instead of an "abelian group".
    • CommentAuthorBCnrd
    • CommentTimeOct 26th 2010
     
    Dear Kevin Lin: Your final comment about schemes over Spec(Z) seems entirely consistent with the description by Andy Putman (not Putnam, btw) of the situations where he'd be inclined to say "Z-algebra". So it appears that everyone's on the same wavelength.
    • CommentAuthorKevin Lin
    • CommentTimeOct 26th 2010
     

    Oops! Fixed. Sorry, Andy! The Putnam/Putman error strikes again.

    • CommentAuthorAndy Putman
    • CommentTimeOct 26th 2010 edited
     
    @Kevin : No problem! I'm pretty resigned to it at this point. When I arrived at Rice several months ago, I found "Putnam" on my office door...

    ps : Alas, we already have a fantastic person organizing the Putnam exam here -- if we didn't, I'd have no choice but to do it just for the comedy value...
    • CommentAuthorEmerton
    • CommentTimeOct 26th 2010
     

    Dear Kevin,

    I can think of at least one interesting map from a Lie algebra to an associative algebra (!) (but it does not take the Lie bracket to the associative multiplication, which is I guess what you meant).

    I agree that the category of Lie algebras and the category of associative algebras don't seem to interact sensibly in the larger category of non-associative algebras; rather their interaction is via enveloping algebras and the like (which is what I was alluding to in the preceding paragraph), and so they seem to be most naturally thought of as living in distinct categories, related by appropriate adjoint functors. This is borne out geometrically, in the way that commutative algebras become rings of functions, Lie algebras becomes spaces of vector fields, and enveloping algebras (as one example of associative but non-commutative algebras) become rings of differential operators.

    Best wishes,

    Matt

    • CommentAuthorHarry Gindi
    • CommentTimeOct 27th 2010 edited
     

    However, in other contexts it would be silly. For instance, I do quite a bit of group theory, and I think I would be laughed at if I called the abelianization of a group a "Z-module" instead of an "abelian group".

    I actually primarily use the term "Z-module" when I write, although I agree that doing so in the context of group theory would sound substantially worse than, say, doing so in the context of commutative algebra.

    Also, I just saw that you made this comment on another thread:

    Wow BCnrd! Aside from the sheer joy of attacking an ant with a bazooka, is there any larger lesson in that proof?

    I just wanted to let you know that I really appreciate the imagery there, and I would like to thank you for making me giggle like a schoolgirl.

  31.  
    A similar question (already asked in MO somewhere) is: Why do recent texts include a 1 in the definition of ring, when there are some interesting examples of ring without 1 ...
  32.  
    Aah---Gerald getting back to the point! But I think the answer is clear. I'm currently lecturing undergraduate ring theory so I had to decide which axioms to put in and which to leave out, and you want to make choices in your original definition of "ring" so as to minimise labour later. So, for example, if I had not put associativity in as an axiom then I think 95 percent of the lemmas/propositions/theorems of the course would start "Let R be an associative ring. Then..." rather than "Let R be a ring. Then...". Perhaps if there were lots of examples of non-associative algebras close to my heart, which then would affect the results I want to lecture about, I would have made another decision. But I want to talk about PIDs and UFDs, so almost made every ring commutative :-)
    • CommentAuthorBCnrd
    • CommentTimeOct 27th 2010
     
    Dear Kevin: In such teaching situations, I make the passing remark to the students that just as matrices and study of various kinds of operators provide useful examples where dropping commutativity is natural, dropping associativity comes up in other important situations. And then I toss in the phrase "Lie algebra"...or maybe I should say "Lie ring"? :)
    • CommentAuthorAndrewL
    • CommentTimeOct 27th 2010
     

    Thank you for all your feedback here.

    First of,on a different issue-but one for which I don't want to open a new thread here-I deleted my post at the thread at MO regarding one line descriptions of mathematical subjects. Things were becoming heated,so I eliminated the problem before it got me suspended for a month.

    Second,the definition of a Lie ring appears clearly in Richard Schafer's AN INTRODUCTION TO NONASSOCIATIVE ALGEBRAS. This short monograph made a big impression on me. The language may have changed among algebraicists since it was written,but from what I've been able to glean from Kevin McCrimmon's work,I don't believe so.

    Thirdly,it may be that the structure of nonassociative rings is too simple to be of much use in algebra. Whether or not it has a more significant role in geometry,I'm not certain.

    • CommentAuthorEmerton
    • CommentTimeOct 27th 2010
     

    Dear AndrewL,

    It is not a question of "too simple" or not. Lie algebras are the most important general example of non-associate algebras, and they play an enormously important role in vast swathes of mathematics, from mathematical physics to number theory (and certainly geometry). But their theory is quite different to the theory of associative algebras. Indeed, the theory of associative, but non-commutative, rings is in turn quite different from the theory of commutative rings. This is why (for example) Lie theory, Wedderburn theory, and commutative algebra are normally treated as distinct subjects.

    Regards,

    Matthew

  36.  

    Actually, the theory of Lie algebras is a subset of the theory of associative rings (at least if you're willing to include Hopf algebras), since the universal enveloping algebra of a Lie algebra actually contains basically all information about it. This is what makes the existence of a separate top level AMS classification for "Nonassociative rings and algebras" so idiotic.

  37.  

    @Ben: as a nonexpert on Lie algebras, I'm curious: is there a precise statement of "basically all information about it"? Perhaps some kind of categorical equivalence?

    • CommentAuthorHarry Gindi
    • CommentTimeOct 27th 2010 edited
     

    @Pete: If I had to guess, I would think that it's something like Morita equivalence. Either that or he's just saying that the universal enveloping algebra functor is full and faithful (something which I'm not actually sure of).

    • CommentAuthorEmerton
    • CommentTimeOct 28th 2010
     

    Dear Pete and Harry,

    Certainly modules for the enveloping algebra are the same as modules for the Lie algebra (just as modules for the group ring of a group are the same as modules for the group). But also, if you include the Hopf algebra structure on the enveloping algebra, then the Lie algebra can be recovered as the subspace of primitive elements.

    Best wishes,

    Matt

  40.  

    @Harry and Matt,

    Yes, I had known that the universal enveloping algebra gives an equivalence of module categories, but that's not as strong as I had wanted. The statement about the Hopf algebra structure certainly satisfies me though. Thanks.

  41.  
    I assume the result Ben was thinking of is: Let g be a Lie algebra over a field of characteristic zero. Let U be the universal enveloping algebra and Delta the coproduct U \to U \otimes U. Then g can be recovered from (Delta, U) as the set of primitive elements: An element f of U is in g if and only if Delta(f) = f \otimes 1 + 1 \otimes f.

    Note that, in characteristic p, if f is primitive then so is f^p, so this result does not hold. However, I believe that there is some more complicated recipe so that g can still be recovered from (Delta, U) in characteristic p.

    Actually, that makes me curious: Can we have two nonisomorphic Lie algebras with isomorphic enveloping algebras, if we ignore the Hopf structure?
  42.  
    Oops, didn't realize that Matt Emerton had already answered this.
    • CommentAuthorWill Jagy
    • CommentTimeOct 28th 2010
     
    They also have those Jordan algebras, like the almonds.

    http://zakuski.math.utsa.edu/~kap/superalgebra.html
  44.  

    David- by the way, this exact point is responsible for the fact that there are two notions of Lie algebra over characteristic p; the obvious one, and a "restricted Lie algebra" which has a operation usually denoted $f\mapsto f^{[p]}$. Then the restricted UEA of a restricted Lie algebra identifies f^p (in the usual multiplication) and f^{[p]}. This is a better notion in many contexts since, as you note, natural Lie algebras that show up, like primitive elements in a Hopf algebra or derivations of a ring have a natural restricted structure over characteristic p.

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