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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.


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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
@Raphael Funny how you started with a ring and ended with a group :). – Srivatsan Sep 2 '11 at 23:09
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
@yoyo: yes, it is true, but for me in particular, is interesting to know the origin of this words. I think that all of us should be know a little about that. – leo Sep 3 '11 at 2:03
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
up vote 149 down vote accepted

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 However, as plausible as that may seem, I don't recall the existence of any historical documents that provide solid evidence in support of such speculations.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
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
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
Bravo for including 'original literature' – Simon S Jul 17 '15 at 11:50

A ring in the mathematical sense is a set S together with two binary operators + and * (commonly interpreted as addition and multiplication, respectively) satisfying the following conditions:

  1. Additive associativity: For all $a,b,c$ in $S$: $(a+b)+c=a+(b+c)$,

  2. Additive commutativity: For all $a,b$ in $S$: $a+b=b+a$,

  3. Additive identity: There exists an element $0$ in $S$ such that for all a in $S$: $0+a=a+0=a$,

  4. Additive inverse: For every $a$ in $S$ there exists $-a$ in $S$ such that $a+(-a)=(-a)+a=0$,

  5. Left and right distributivity: For all $a,b,c$ in $S$: $a*(b+c)=(a*b)+(a*c)$ and $(b+c)*a=(b*a)+(c*a)$,

  6. Multiplicative associativity: For all $a,b,c$ in $S$, $(a*b)*c=a*(b*c)$ (a ring satisfying this property is sometimes explicitly termed an associative ring).

Conditions 1-5 are always required. Though non-associative rings exist, virtually all texts also require condition 6 (Itô 1986, pp. 1369-1372; p. 418; Zwillinger 1995, pp. 141-143; Harris and Stocker 1998; Knuth 1998; Korn and Korn 2000; Bronshtein and Semendyayev 2004).

Rings may also satisfy various optional conditions:

  1. Multiplicative commutativity: For all $a,b$ in $S$, $a*b=b*a$ (a ring satisfying this property is termed a commutative ring),

  2. Multiplicative identity: There exists an element $1$ in $S$ such that for all $a\neq0$ in $S$: $1*a=a*1=a$ (a ring satisfying this property is termed a unit ring, or sometimes a "ring with identity"),

  3. Multiplicative inverse: For each $a\neq0$ in $S$, there exists an element $a^{-1}$ in $S$ such that for all $a\neq0$ in $S$, $a*a^{-1}=a^{-1}*a=1$, where $1$ is the identity element.

A ring satisfying all additional properties 6-9 is called a field, whereas one satisfying only additional properties 6, 8, and 9 is called a division algebra (or skew field).

Some authors depart from the normal convention and require (under their definition) a ring to include additional properties. For example, Birkhoff and Mac Lane (1996) define a ring to have a multiplicative identity (i.e., property 8).

Here are a number of examples of rings lacking particular conditions:

  1. Without multiplicative associativity (sometimes also called nonassociative algebras): octonions, OEIS A037292,

  2. Without multiplicative commutativity: Real-valued $2×2$ matrices, quaternions,

  3. Without multiplicative identity: Even-valued integers,

  4. Without multiplicative inverse: integers.

The word ring is short for the German word 'Zahlring' (number ring). The French word for a ring is anneau, and the modern German word is Ring, both meaning (not so surprisingly) "ring." Fraenkel (1914) gave the first abstract definition of the ring, although this work did not have much impact. The term was introduced by Hilbert to describe rings like $\mathbb{Z}[\sqrt[3]{2}]=\{a+b\sqrt[3]{2}+c\sqrt[3]{4} \mid a,b,c \in \mathbb{Z}\}$.

By successively multiplying the new element $\sqrt[3]{2}$, it eventually loops around to become something already generated, something like a ring, that is, $\sqrt[3]{2}^2=\sqrt[3]{4}$ is new but $\sqrt[3]{2}^3=2$ is an integer. All algebraic numbers have this property, e.g., $\eta=\sqrt(2)+\sqrt(3)$ satisfies $\eta^4=10\eta^2-1$.

A ring must contain at least one element, but need not contain a multiplicative identity or be commutative. The number of finite rings of n elements for $n=1, 2, ...$, are $1, 2, 2, 11, 2, 4, 2, 52, 11, 4, 2, 22, 2, 4, 4, ...$ (OEIS A027623 and A037234; Fletcher 1980). If p and q are prime, there are two rings of size $p$, four rings of size $pq$, 11 rings of size $p^2$ (Singmaster 1964, Dresden), 22 rings of size $p^2q$, 52 rings of size $p^3$ for $p=2$, and 53 rings of size $p^3$ for $p>2$ (Ballieu 1947, Gilmer and Mott 1973; Dresden).

A ring that is commutative under multiplication, has a unit element, and has no divisors of zero is called an integral domain. A ring whose nonzero elements form a commutative multiplication group is called a field. The simplest rings are the integers $\mathbb{Z}$, polynomials $\mathbb{R}[x]$ and $\mathbb{R}[x,y]$ in one and two variables, and square $n×n$ real matrices.

Rings which have been investigated and found to be of interest are usually named after one or more of their investigators. This practice unfortunately leads to names which give very little insight into the relevant properties of the associated rings.

Renteln and Dundes (2005) give the following (bad) mathematical joke about rings:

Q: What's an Abelian group under addition, closed, associative, distributive, and bears a curse? A: The Ring of the Nibelung. SEE ALSO: Abelian Group, Artinian Ring, Chow Ring, Dedekind Ring, Division Algebra, Endomorphism Ring, Field, Gorenstein Ring, Group, Group Ring, Ideal, Integral Domain, Module, Nilpotent Element, Noetherian Ring, Noncommutative Ring, Number Field, Prime Ring, Prüfer Ring, Quotient Ring, Regular Ring, Ring of Integers, Ringoid, Semiprime Ring, Semiring, Semisimple Ring, Simple Ring, Trivial Ring, Unit Ring, Zero Divisor REFERENCES:

Allenby, R. B. Rings, Fields, and Groups: An Introduction to Abstract Algebra, 2nd ed. Oxford, England: Oxford University Press, 1991.

Ballieu, R. "Anneaux finis; systèmes hypercomplexes de rang trois sur un corps commutatif." Ann. Soc. Sci. Bruxelles. Sér. I 61, 222-227, 1947.

Beachy, J. A. Introductory Lectures on Rings and Modules. Cambridge, England: Cambridge University Press, 1999.

Berrick, A. J. and Keating, M. E. An Introduction to Rings and Modules with K-Theory in View. Cambridge, England: Cambridge University Press, 2000.

Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 5th ed. New York: Macmillian, 1996.

Bronshtein, I. N.; Semendyayev, K. A.; Musiol, G.; and Muehlig, H. Handbook of Mathematics, 4th ed. New York: Springer-Verlag, 2004.

Dresden, G. "Small Rings."

Ellis, G. Rings and Fields. Oxford, England: Oxford University Press, 1993.

Fine, B. "Classification of Finite Rings of Order $p^2$." Math. Mag. 66, 248-252, 1993.

Fletcher, C. R. "Rings of Small Order." Math. Gaz. 64, 9-22, 1980.

Fraenkel, A. "Über die Teiler der Null und die Zerlegung von Ringen." J. reine angew. Math. 145, 139-176, 1914.

Gilmer, R. and Mott, J. "Associative Rings of Order $p^3$." Proc. Japan Acad. 49, 795-799, 1973.

Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, 1998.

Itô, K. (Ed.). "Rings." §368 in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 2. Cambridge, MA: MIT Press, 1986.

Kleiner, I. "The Genesis of the Abstract Ring Concept." Amer. Math. Monthly 103, 417-424, 1996.

Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addison-Wesley, 1998.

Korn, G. A. and Korn, T. M. Mathematical Handbook for Scientists and Engineers. New York: Dover, 2000.

Nagell, T. "Moduls, Rings, and Fields." §6 in Introduction to Number Theory. New York: Wiley, pp. 19-21, 1951.

Renteln, P. and Dundes, A. "Foolproof: A Sampling of Mathematical Folk Humor." Notices Amer. Math. Soc. 52, 24-34, 2005.

Singmaster, D. and Bloom, D. M. "Problem E1648." Amer. Math. Monthly 71, 918-920, 1964.

Sloane, N. J. A. Sequences A027623 and A037234 in "The On-Line Encyclopedia of Integer Sequences."

van der Waerden, B. L. A History of Algebra. New York: Springer-Verlag, 1985.

Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1168, 2002.

Zwillinger, D. (Ed.). "Rings." §2.6.3 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 141-143, 1995. Referenced on Wolfram|Alpha: Ring

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98% of this post has nothing to do with the question asked. – Greg Martin 8 hours ago
Aside from the fact that this has been plagiarised from Wolfram Mathworld, the vast majority of what has been written does not answer the question, and the single sentence of relevant information is already in Bill's answer (posted $5$ years ago). The fact that this has gotten $5$ upvotes in half an hour is astounding. – MathematicsStudent1122 8 hours ago

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