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at the beginning of ring I have spelled out a more explicit definition. Also added the examples of rings on cyclic groups to explain the origin of the word “ring”.
to explain the origin of the word “ring”.
I didn’t know that! Do you have a source for that?
Do you have a source for that?
Hm. Let’s see. That’s what they told me when I was a gullible student. I never checked the originals. The entry Mathworld – Ring tells its readers the same story:
The term was introduced by Hilbert to describe rings like […] By successively multiplying the new element […], it eventually loops around to become something already generated, something like a ring,
but apparently it’s just a story, not a review of Hilbert’s way of introducing the term.
Okay, so I went to Google books and read Hilbert’s original article
David Hilbert, Die Theorie der algebraischen Zahlkörper, Jahresbericht der Deutschen Mathematiker-Vereinigung 4 (1879)
and there, in section 9.31, indeed no motivation like this is given. Instead it just says:
[…] ein Zahlring, Ring oder Integritätsbereich genannt
with a footnote that reads
Nach Dedekind “eine Ordnung”.
And that’s it. But so that means already that I was wrong, since neither Dedekind nor Hilbert meant to invoke the picture of clock arithmetic. (And Hilbert does not even mention anything as simple as $\mathbb{Z}_n$).
Now I don’t have Dedekind’s original text. Because also “Ordnung” is ambiguous. One meaning is “order” as in “ordered set”. But it also is used in the sense of “a collection of beings of the same nature” in the sense used as a taxonomic rank.
Dedekind can’t have meant “ordered set”. So he must have meant “taxonomical order”. Maybe thinking of a “taxonomy of numbers”? I don’t don’t know.
But anyway, I suppose that Hilbert’s “ring” is therefore also to be read as meaning “collection of beings”, as in “drug-dealer ring”. :-)
Which, I must say, is too bad.
What is the Dedekind’s original text ?
I don’t know, I just meant to say that I haven’t seen any original text on Dedekind’s “Ordnungen”. Maybe he didn’t even write it up. He is just being credited for the idea (by Hilbert).
I have made further notes at ring - References - History
A google search did turn up this, but that does not seem to answer the question.
A google search did turn up this,
Yes, that’s already linked to in the entry.
but that does not seem to answer the question.
I believe I just answered the question in #3. :-)
By the way, I had put some text into the Idea-section. Not meant to be perfect. Please edit as you see the need.
There is an Stackexchange question here, with another historical reference.
I added the following standard observation
The structure of an $A\otimes A^{op}$-ring $(R,\mu_R,\eta)$ is determined by the structure of $A$ as a ring, together with the two natural homomorphisms of rings $s = \eta(-\otimes 1_A):A\to R$ and $t=\eta(1_A\otimes -):A^{op}\to R$ which have commuting images ($s(a)t(a')=t(a')s(a)$, for all $a,a'\in A$).
This is very interesting when dualizing the notion of groupoid (algebra of functions/space duality) – source and target map in algebraic language get sometimes conveniently packed into $A\otimes A^{op}$-ring language, as in the case of bialgebroids.
You should put something around that paragraph, wrapping it, something that allows to discern it as a new idea within the text that surropunds it. At least maybe a remark-environment.
I knew it had to be there, but even so I only found it after hitting see changes.
In this article there is a remark which says that for unital rings commutativity of addition is not a needed axiom, but it’s necessary for nonunital rings.
I completed this remark with the following.
I’ve added a case of nonunital rings in which this axiom is not needed. The proof is pretty much the same as for the unital rings. Those are nonunital rings of which multiplicative semigroup is left/right weakly reductive i. e. $x\cdot a = x\cdot b$ for all $x$ implies $a = b$, or the same from the other side.
I wrote a brief (and probably a bit messy) article here on nlab about such semigroups if you need more information.
The only purpose of unity here (here meaning - this remark) is so that the multiplication can distinguish between the elements of our multiplicative semigroup. Of course this is the most basic property one would desire. Another property like this is left/right weak reduction.
Perhaps someone thinks that I should’ve written the weak reductive property more explicitly in the article, but I didn’t want to digress any more from the subject which are rings.
Adam
I am more familiar with ’cancellative’ than I am with ’reductive’. See cancellative monoid.
Weak reductive isn’t the same as cancellative, although the latter implies the former. What makes the two different is placement of quantifiers.
Note that in this case it doesn’t even make sense for the multiplicative semigroup to be cancellative, as 0 is never going to be cancelled. Instead, we can demand that there exists an element which can be cancelled from one of the sides.
Adam
.
Added a link to weakly reductive semigroup.
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