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I've done some search in Internet and other sources about this question. Why the name ring to this particular object? Just curiosity.

Thanks.

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In German, "Ring" can also mean a (close) group of people with shared interests, an association. Maybe it is because of that: you've got some closely interacting elements but for all their interaction, they never leave the group. –  Raphael Sep 2 '11 at 23:05
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@Raphael Funny how you started with a ring and ended with a group :). –  Srivatsan Sep 2 '11 at 23:09
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ring group field manifold etc are just general words referring to collections of things that were "jargonized" –  yoyo Sep 2 '11 at 23:14
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Did nobody thought that maybe -- just maybe -- that perhaps the bloke who coined the name thought the name just sounds cool? I mean, if you had the chance to coin a name for something you discovered, wouldn't some of you want to name it Diamond or something like that? –  Lie Ryan Sep 3 '11 at 1:43
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leo: I totally agree with your last comment. It goes hand in hand with the "read the masters" credo to know and try to find out a little about where the words we use on a daily basis come from. While in principle, as Hilbert allegedly put it, "One must be able to say at all times--instead of points, straight lines, and planes--tables, chairs, and beer mugs", I think knowing the etymology of words and the history of ideas is part of the general culture a mathematician should have. –  t.b. Sep 3 '11 at 3:02
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1 Answer

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The name "ring" is derived from Hilbert's term "Zahlring" (number ring), introduced in his Zahlbericht for certain rings of algebraic integers. As for why Hilbert chose the name "ring", I recall reading speculations that it may have to do with cyclical (ring-shaped) behavior of powers of algebraic integers. Namely, if $\:\alpha\:$ is an algebraic integer of degree $\rm\:n\:$ then $\:\alpha^n\:$ is a $\rm\:\mathbb Z$-linear combination of lower powers of $\rm\:\alpha\:,\:$ thus so too are all higher powers of $\rm\:\alpha\:.\:$ Hence all powers cycle back onto $\rm\:1,\:\alpha,\:,\ldots,\alpha^{n-1}\:,\:$ i.e. $\rm\:\mathbb Z[\alpha]\:$ is a finitely generated $\:\mathbb Z$-module. Possibly also the motivation for the name had to do more specifically with rings of cyclotomic integers. However, as plausible as that may seem, I don't recall the existence of any historical documents that provide solid evidence in support of such speculations.

Beware that one has to be very careful when reading such older literature. Some authors mistakenly read modern notions into terms which have no such denotation in their original usage. To provide some context I recommend reading Lemmermeyer and Schappacher's Introduction to the English Edition of Hilbert’s Zahlbericht. Below is a pertinent excerpt.

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Below is an excerpt from Leo Corry's Modern algebra and the rise of mathematical structures, p. 149.

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Below are a couple typical examples of said speculative etymology of the term "ring" via the "circling back" nature of integral dependence, from Harvey Cohn's Advanced Number Theory, p. 49.

$\quad$The designation of the letter $\mathfrak D$ for the integral domain has some historical importance going back to Gauss's work on quadratic forms. Gauss $\left(1800\right)$ noted that for certain quadratic forms $Ax^2+Bxy+Cy^2$ the discriminant need not be square-free, although $A$, $B$, $C$ are relatively prime. For example, $x^2-45y^2$ has $D=4\cdot45$. The $4$ was ignored for the reason that $4|D$ necessarily by virtue of Gauss's requirement that $B$ be even, but the factor of $3^2$ in $D$ caused Gauss to refer to the form as one of "order $3$." Eventually, the forms corresponding to a value of $D$ were called an "order" (Ordnung). Dedekind retained this word for what is here called an "integral domain."

$\quad$The term "ring" is a contraction of "Zahlring" introduced by Hilbert $\left(1892\right)$ to denote (in our present context) the ring generated by the rational integers and a quadratic integer $\eta$ defined by $$\eta^2+B\eta+C=0.$$ It would seem that module $\left[1,\eta\right]$ is called a Zahlring because $\eta^2$ equals $-B\eta-C$ "circling directly back" to an element of $\left[1,\eta\right]$ . This word has been maintained today. Incidentally, every Zahlring is an integral domain and the converse is true for quadratic fields.

and from Rotman's Advanced Modern Algebra, p. 81.

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"Beware that one has to be very careful when reading such older literature. Some authors mistakenly read modern notions into terms which have no such denotation in their original usage." deserves a +1 on its own. –  J. M. Sep 3 '11 at 1:31
    
Thank you very much @Bill. –  leo Sep 3 '11 at 2:05
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Bill, I think it would be worth pointing readers to Kleiner's article From Numbers to Rings: The Early History of Ring Theory which you mentioned here. –  t.b. Sep 3 '11 at 3:15
    
@Theo Kleiner's article is a nice place to start to learn about the history of ring theory. But it does not explicitly address the OP's question (unless I missed something there). –  Bill Dubuque Sep 3 '11 at 3:47
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I don't think you missed anything. I just thought that people interested in the etymology of the word "ring" will likely be interested in the history of rings, and that's why I pointed to your other answer. –  t.b. Sep 3 '11 at 3:57
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