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I keep reading questions like: "prove that this ring has a unit element under such axioms" or "prove that morphisms of rings preserve the unit element under those circumstances"
I find these questions weird because I would have thought that any moderately recent document (textbook, article, ...) incorporates those requirements on units in the definition of ring and ring morphism, implicitly or explicitly.

First question: Are there any counterexamples, for example authoritative recent ($\leq 25$ years) textbooks defining rings without requiring unit elements ? Or do I suffer from the diseases bourbakitis and algebrogeometritis?

Second question: Should we tell questioners that their terminology is not optimal (in a sociological , not a logical sense of course) and advise them to use Jacobson's happy terminology "rngs" in the absence of unit, or point out that their "rings" are actually ideals of genuine rings.
That said, answerers would of course be encouraged to answer the OP's question, after this little terminological admonition.

Of course, you might say: "Do as you like when you answer and don't bother us, Georges"
This is indeed excellent advice, but the point is that I tend not to answer these questions (I concentrate on other tags) and nevertheless would like users to encourage the usage I advocate if indeed they agree.

Edit: conclusion
Well, my question has been answered : yes, some very active users here defend the definition in which rings may not have a unity .
So that we need no common policy on answers to related questions, and I suppose we will all go on using the definition adopted by the mathematical tribe we belong to: no harm in that!
Thanks to you all for your contributions .

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    $\begingroup$ Welp, one thing to do is to at least tag those questions (rngs) if they aren't already. I think when that tag is present context will be enough to suggest certain definitions over others. $\endgroup$
    – Willie Wong
    Apr 4, 2012 at 12:33
  • $\begingroup$ Dear @Willie: that's a really great suggestion! (I didn't even know such a tag existed...) $\endgroup$
    – Georges Elencwajg
    Apr 4, 2012 at 12:40
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    $\begingroup$ I have two colleagues that would violently disagree with your assertion that using "rings" to denote what Jacobson calls 'rngs' is "not optimal", and they seem to keep rather large company, including distinguished ring theorists. They continue to publish quite a bit. I will also note that Dummit and Foote don't require rings to have unity. $\endgroup$
    – Arturo Magidin
    Apr 4, 2012 at 13:35
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    $\begingroup$ Neither does Herstein. Where I study, calling rngs "rings" is a very common practice, especially for radical theorists. Not requiring associativity also happens. $\endgroup$
    – user23211
    Apr 4, 2012 at 15:07
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    $\begingroup$ @GeorgesElencwajg: I just meant that my two colleagues work in rings without unity, and never bat an eye at calling them "rings". They specify when they want their rings to have unity, and firmly believe that you have "rings" and "rings with unity", and not "rings" and "rings possibly without unity/rngs". They're both accomplished researches, and have no problems with unities, they just don't like being told that rings "should" have unities, and are very vocal about that. $\endgroup$
    – Arturo Magidin
    Apr 4, 2012 at 15:07
  • $\begingroup$ Dear @Arturo: yes , Dummit and Foot do not require rings to have a unity. However they do require that integral domains have a unity. And what I find strange is that they only define polynomial rings over rings that do have a unity (in section 7.2). They also have blanket assumptions that all rings have unity in for example sections 7.4, 7.6, all of chapters 15, 16, ... By the way, could you persuade your two colleagues to explain their violent disagreement here: I find this very interesting (and disagreement is what I was looking for with my question!) $\endgroup$
    – Georges Elencwajg
    Apr 4, 2012 at 15:14
  • $\begingroup$ Another data point: "Cogroups and Co-rings in Categories of Associative Algebras" by George M. Bergman and Adam O. Hausknecht uses $\mathbf{Ring}$ for the variety of rings, not necessarily with unity (and morphisms are not required to respect unity when they do have it), and $\mathbf{Ring}^1$ for the variety of unital rings. Published in 1996. Disclaimer: Bergman doesn't like the term "monoid", saying we should not multiply nomenclature without need, and prefers "semigroup with identity." He probably objects to introducing a new term for "rings without $1$" on principle. $\endgroup$
    – Arturo Magidin
    Apr 4, 2012 at 15:17
  • $\begingroup$ Dear @arturo: oops, our comments crossed each other because I modified mine. But of course I agree with your colleagues: they are perfectly entitled to use their terminology . And you are perfectly right to tell me about them, since this what I was asking about:r: thanks a lot! $\endgroup$
    – Georges Elencwajg
    Apr 4, 2012 at 15:20
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    $\begingroup$ Dear @Arturo, why not write an answer to the question, since you have obviously much to say? Your information will then have more impact than in comments. $\endgroup$
    – Georges Elencwajg
    Apr 4, 2012 at 15:22
  • $\begingroup$ @GeorgesElencwajg: I'll get back to this a bit later and add more info when I get a chance to look through my books. There's something I need to look up in Bergman's Universal Algebra book... $\endgroup$
    – Arturo Magidin
    Apr 4, 2012 at 15:24
  • $\begingroup$ Dear @ymar: yes I knew about Herstein and that is why I added the condition "$\leq 25$ years". Not requiring associativity for rings is a bit stiff, though! I think people usually tend to speak of algebras in that case. $\endgroup$
    – Georges Elencwajg
    Apr 4, 2012 at 15:28
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    $\begingroup$ Wikipedia had an extensive discussion four years ago including a table of publications $\endgroup$
    – Henry
    Apr 6, 2012 at 22:43
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    $\begingroup$ I think this question should better fit the main MSE site, rather than meta. $\endgroup$
    – leonbloy
    Apr 7, 2012 at 21:39
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    $\begingroup$ @leonbloy I flagged it for mods to move to main, but an anonymous mod declined to do so. $\endgroup$
    – Bill Dubuque
    May 1, 2012 at 19:32
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    $\begingroup$ I'm looking at a textbook, published in 2015, in which rings are specifically defined to have a multiplicative identity, and then a few pages later there is an exercise: Are the rings $\bf Z$ and $2{\bf Z}$ isomorphic? $\endgroup$
    – Gerry Myerson
    Nov 26, 2017 at 0:49

2 Answers 2

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There is a tradition in measure theory to call rngs satisfying $a^2 = a$ Boolean rings while Boolean algebras are those having a unit. This can be seen for example in Fremlin's text measure theory Volume 3, chapter 31 from the last decade.

However, I admit that this is a rare exception.

Much more common seem to be algebras that can have a unit or not; This is even in line with Bourbaki, who calls an algebra with unit algèbre unifère. Those arise prominently in functional analytic contexts, where many of the examples simply don't come naturally equipped with a unit and the standard unitization from algebra does not give particularly interesting objects: for example, one can choose to look at the continuous functions on any compactification of a locally compact Hausdorff space $X$ to get a unitization of the ring of continuous functions $X$ vanishing at infinity and I hope everyone agrees that there are much more interesting and useful compactifications than the one-point compactification.

All this can probably be dismissed as a clash of cultures between analysts and algebraists.

I do like to think of algebras as a special case of rings, so this could be taken as an argument for not insisting that rings have units. On the other hand, I tend to agree with the points raised in the nice blurb of Keith Conrad on the definition of rings.

I guess we just have to live with the fact that some fairly popular algebra books like Herstein's have rings without unit and I think it is worthwhile to point out to less experienced users of the site that the overwhelming majority of people tacitly assume the existence of a unit whenever they hear the word ring. Most of the time people take the effort to point out explicitly that they do not insist on a unit in a ring if they happen to need or want to do so.

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    $\begingroup$ +1: nothing to add to or subtract from this thoughtful, well-argumented , balanced answer. $\endgroup$
    – Georges Elencwajg
    Apr 4, 2012 at 13:24
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    $\begingroup$ Yes. In measure theory, they talk about rings of sets, and algebras of sets. It is an algebra iff the whole space is in it. The whole space is the unit. $\endgroup$
    – GEdgar
    Apr 5, 2012 at 12:19
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Non unital rings and algebras are necessary when discussing crossed modules of rings and algebras. To relate to another point in the meta, crossed modules are also equivalent to internal groupoids in the categories of these objects; and since groupoids generalise equivalence relations, crossed modules are also related to quotient constructions!

Relevant to this is that it is convenient to think of an ideal $I$ in a ring $R$, usually itself unital, as a subring with an operation of $R$ on the left and right of $I$. This generalises to having a ring morphism $\mu: M \to R$ (with image an ideal) together with operations of $R$ on the left and right of $M$ satisfying some obvious axioms and also such that $mn=\mu(m)n =m \mu(n)$ for all $m,n \in M$.

And of course if an ideal $I$ contains the unit of $R$ then it coincides with $R$.

Another extension of terminology is I think called a rig, in which the addition need not have negatives.

I hope that helps.

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    $\begingroup$ But rigs are also called semirings (and I think more commonly so, by people that work with them). $\endgroup$
    – Marc van Leeuwen
    Sep 2, 2015 at 9:55

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