Editing Ring (mathematics) (section)

== History ==
{{See also|Ring theory#History}}
[[File:Dedekind.jpeg|thumb|100px|right|[[Richard Dedekind]], one of the founders of [[ring theory]]]]

=== Dedekind ===
The study of rings originated from the theory of [[polynomial ring]]s and the theory of [[algebraic integer]]s.<ref name="history">{{cite web| url = https://mathshistory.st-andrews.ac.uk/HistTopics/Ring_theory/| title = The development of Ring Theory}}</ref> In 1871, [[Richard Dedekind]] defined the concept of the ring of integers of a number field.{{sfnp|Kleiner|1998|p=27|ps=}} In this context, he introduced the terms "ideal" (inspired by [[Ernst Kummer]]'s notion of ideal number) and "module" and studied their properties. Dedekind did not use the term "ring" and did not define the concept of a ring in a general setting.

=== Hilbert ===
The term "Zahlring" (number ring) was coined by [[David Hilbert]] in 1892 and published in 1897.{{sfnp|Hilbert|1897|ps=}} In 19th century German, the word "Ring" could mean "association", which is still used today in English in a limited sense (for example, spy ring),{{CN|date=October 2023}} so if that were the etymology then it would be similar to the way "group" entered mathematics by being a non-technical word for "collection of related things". According to Harvey Cohn, Hilbert used the term for a ring that had the property of "circling directly back" to an element of itself (in the sense of an [[Equivalence relation|equivalence]]).{{sfnp|Cohn|1980|loc=[https://archive.org/details/advancednumberth00cohn_0/page/49 p. 49]|ps=}} Specifically, in a ring of algebraic integers, all high powers of an algebraic integer can be written as an integral combination of a fixed set of lower powers, and thus the powers "cycle back". For instance, if {{math|1=''a''{{sup|3}} − 4''a'' + 1 = 0}} then:
:<math>\begin{align}
a^3 &= 4a-1, \\
a^4 &= 4a^2-a, \\ 
a^5 &= -a^2+16a-4, \\ 
a^6 &= 16a^2-8a+1, \\
a^7 &= -8a^2+65a-16, \\
\vdots \ & \qquad \vdots
\end{align}</math>
and so on; in general, {{math|''a''{{sup|''n''}}}} is going to be an integral linear combination of {{math|1}}, {{math|''a''}}, and {{math|''a''{{sup|2}}}}.

=== Fraenkel and Noether ===
The first axiomatic definition of a ring was given by [[Abraham Fraenkel|Adolf Fraenkel]] in 1915,{{sfnp|Fraenkel|1915|pp=143–145|ps=}}{{sfnp|Jacobson|2009|p=86|loc=footnote 1|ps=}} but his axioms were stricter than those in the modern definition. For instance, he required every [[zero divisor|non-zero-divisor]] to have a [[multiplicative inverse]].{{sfnp|Fraenkel|1915|p=144|loc=axiom ''R''{{sub|8)}}|ps=}} In 1921, [[Emmy Noether]] gave a modern axiomatic definition of commutative rings (with and without 1) and developed the foundations of commutative ring theory in her paper ''Idealtheorie in Ringbereichen''.{{sfnp|Noether|1921|p=29|ps=}}

=== Multiplicative identity and the term "ring" ===
Fraenkel applied the term "ring" to structures with axioms that included a multiplicative identity,{{sfnp|Fraenkel|1915|p=144|loc=axiom ''R''{{sub|7)}}|ps=}} whereas Noether applied it to structures that did not.{{sfnp|Noether|1921|p=29|ps=}}

Most or all books on algebra{{sfnp|van der Waerden|1930|ps=}}{{sfnp|Zariski|Samuel|1958|ps=}} up to around 1960 followed Noether's convention of not requiring a {{math|1}} for a "ring". Starting in the 1960s, it became increasingly common to see books including the existence of {{math|1}} in the definition of "ring", especially in advanced books by notable authors such as Artin,{{sfnp|Artin|2018|p=346|ps=}} Bourbaki,{{sfnp|Bourbaki|1989|p=96|ps=}} Eisenbud,{{sfnp|Eisenbud|1995|p=11|ps=}} and Lang.{{sfnp|Lang|2002|p=83|ps=}} There are also books published as late as 2022 that use the term without the requirement for a {{math|1}}.{{sfnp|Gallian|2006|p=235|ps=}}{{sfnp|Hungerford|1997|p=42|ps=}}{{sfnp|Warner|1965|p=188|ps=}}{{sfnp|Garling|2022|ps=}} Likewise, the [[Encyclopedia of Mathematics]] does not require unit elements in rings.<ref>{{cite web |url=https://www.encyclopediaofmath.org/index.php/Associative_rings_and_algebras |title=Associative rings and algebras |website=Encyclopedia of Mathematics}}</ref> In a research article, the authors often specify which definition of ring they use in the beginning of that article.

Gardner and Wiegandt assert that, when dealing with several objects in the category of rings (as opposed to working with a fixed ring), if one requires all rings to have a {{math|1}}, then some consequences include the lack of existence of infinite direct sums of rings, and that proper direct summands of rings are not subrings. They conclude that "in many, maybe most, branches of ring theory the requirement of the existence of a unity element is not sensible, and therefore unacceptable."{{sfnp|Gardner|Wiegandt|2003|ps=}}  [[Bjorn Poonen|Poonen]] makes the counterargument that the natural notion for rings would be the [[direct product]] rather than the direct sum. However, his main argument is that rings without a multiplicative identity are not totally associative, in the sense that they do not contain the product of any finite sequence of ring elements, including the empty sequence.{{efn|text=Poonen claims that "the natural extension of associativity demands that rings should contain an empty product, so it is natural to require rings to have a&nbsp;{{math|1}}".}}{{sfnp|Poonen|2019|ps=}}

Authors who follow either convention for the use of the term "ring" may use one of the following terms to refer to objects satisfying the other convention:
* to include a requirement for a multiplicative identity: "unital ring", "unitary ring", "unit ring", "ring with unity", "ring with identity", "ring with a unit",{{sfnp|Wilder|1965|p=176|ps=}} or "ring with&nbsp;1".{{sfnp|Rotman|1998|p=7|ps=}}
* to omit a requirement for a multiplicative identity: "rng"{{sfnp|Jacobson|2009|p=155|ps=}} or "pseudo-ring",{{sfnp|Bourbaki|1989|p=98|ps=}} although the latter may be confusing because it also has other meanings.