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\begin{document}

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\section*{ring}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{linear_algebra}{}\paragraph*{{Linear algebra}}\label{linear_algebra}

[[!include homotopy - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{generalizations}{Generalizations}\dotfill \pageref*{generalizations} \linebreak
\noindent\hyperlink{internalising_the_sets}{Internalising the sets}\dotfill \pageref*{internalising_the_sets} \linebreak
\noindent\hyperlink{internalising_the_abelian_groups}{Internalising the abelian groups}\dotfill \pageref*{internalising_the_abelian_groups} \linebreak
\noindent\hyperlink{rings_over_a_ring_rings}{Rings over a ring ($A$-rings)}\dotfill \pageref*{rings_over_a_ring_rings} \linebreak
\noindent\hyperlink{higher_rings}{Higher rings}\dotfill \pageref*{higher_rings} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{types_of_rings}{Types of rings}\dotfill \pageref*{types_of_rings} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak
\noindent\hyperlink{ReferencesGeneral}{General}\dotfill \pageref*{ReferencesGeneral} \linebreak
\noindent\hyperlink{ReferencesHistory}{History}\dotfill \pageref*{ReferencesHistory} \linebreak


\hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea}

A \emph{ring} (also: \emph{number ring}) is a basic structure in [[algebra]]: a [[set]] equipped with two [[binary operation]]s called \emph{addition} and \emph{multiplication}, such that the operation of addition forms an [[abelian group]] and the operation of multiplication a [[monoid]] structure which [[distributivity law|distributes]] over addition.

All the familiar number systems such as the [[integer numbers]], [[rational numbers]], [[real numbers]], [[complex]] numbers are rings under the standard operations of addition and multiplication. Except for the first in this list they are indeed [[fields]], which are rings in which also the multiplication operation has an inverse for every element except 0 (the additive neutral element).

Other basic examples of rings are the [[cyclic groups]] $\mathbb{Z}_n$ under their mod-$n$ operations inherited from the integers ([[cyclic rings]]); the [[polynomial rings]], etc.

More abstractly, a [[ring]] is a [[monoid object|monoid]] [[internalization|internal to]] [[abelian groups]] (with their [[tensor product of abelian groups]]), and this perspective helps to explain the central relevance of the concept, owing to the fundamental nature of the notion of \emph{[[monoid objects]]}. Accordingly monoids internal to other [[abelian categories]] and more generally [[stable infinity-categories]] constitute generalizations of the notion of \emph{ring} that are of interest. Notably when abelian groups are generalized to their analogs in [[stable homotopy theory]], namely to [[spectra]], the corresponding internal monoids are [[E-infinity rings]], a basic structure in [[higher algebra]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{}\hypertarget{}{}
A [[ring]] (unital and not-necessarily commutative) is an [[abelian group]] $R$ equipped with

\begin{enumerate}%
\item an element $1 \in R$


\item a [[bilinear map]], hence a [[group homomorphism]]

\begin{displaymath}
\cdot : R \otimes R \to R
\end{displaymath}
out of the [[tensor product of abelian groups]],



\end{enumerate}
such that $\cdot$ is [[associativity law|associative]] and [[unit law|unital]] with respect to 1.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The fact that the product is a [[bilinear map]] is the \textbf{[[distributivity law]]}: for all $r, r_1, r_2 \in R$ we have

\begin{displaymath}
r \cdot (r_1 + r_2) = r \cdot r_1 + r \cdot r_2
\end{displaymath}
and

\begin{displaymath}
(r_1 + r_2) \cdot r = r_1 \cdot r + r_2 \cdot r
  \,.
\end{displaymath}
\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
A (unital, non-commutative) \textbf{ring} is (equivalently)

\begin{itemize}%
\item a [[monoid object|monoid]] [[internalization|internal to]] [[Ab]] regarded as a [[monoidal category]] equipped with the [[tensor product of abelian groups]];
\item a [[pointed object|pointed]] [[enriched category|category enriched over]] [[Ab]] with a single [[object]].
\item a [[ringoid]] with a single [[object]].

\end{itemize}
A \textbf{commutative} (unital) ring is a [[commutative monoid]] object in $(Ab, \otimes)$.

\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
In usual ring theory people often talk about \textbf{[[nonunital rings]]} as well: multiplicative [[semigroups]] with additive [[abelian group]] structure where the multiplication is distributive toward addition; these are semigroup objects in $Ab$. As in the unital case, if the semigroup is abelian then the ring is said to be \textbf{commutative nonunital}. Note the adjective `nonunital' is an example of the [[red herring principle]].

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
If one removes the assumption that the additive group is abelian but retains the remaining ring axioms, the result is still a ring. (Expand $(1 + a)(1 + b)$ in two different ways, and cancel to conclude $a + b = b + a$.) The result is false for nonunital rings: for any group $(G, +)$ we could define multiplication to be the [[constant function]] at the additive identity, and all the axioms except additive commutativity are trivially satisfied.

\end{remark}
\hypertarget{generalizations}{}\subsection*{{Generalizations}}\label{generalizations}

It is possible to [[internalization|internalise]] the notion of ring in at least two different ways. Either one can replace the [[Set|category of sets]] in the classical definition with another category $C$ -- see [[ring object]] -- , or one can replace [[Ab]] in the fancy definition with another category $M$.

\hypertarget{internalising_the_sets}{}\subsubsection*{{Internalising the sets}}\label{internalising_the_sets}

If $C$ is a [[cartesian monoidal category]], then any [[Lawvere theory]] may be internalised in $C$. The theory of rings is an example, so we can speak of \emph{ring objects} in $C$. Then a ring object in $Set$ is simply a ring. (This works whether your rings are unital or nonunital, commutative or noncommutative, etc.) However, not every notion of internal ring takes this form.

The theory of rings is a combination of a monoid (or semigroup, if nonunital) and an abelian group structure. Thus, ring objects are algebras over a composed [[operad]] (or [[monad]]) of a monoid operad and an abelian group operad, using a standard [[distributive law]] for that situation in the sense of operads (or monads), which corresponds to the usual distributive law in the classical definition of a ring.

A particular example of this is a ring in a [[topos]]. In a topos one usually alternatively defines a ring object by the standard set-theoretic definition of a ring, and interpret the formulas in the sense of topos-theoretic semantics.

Picking a ring object $R$ in a [[topos]] $\mathcal{T}$ promotes it into a [[ringed topos]].

In cartesian categories one can also define the structure of an (abelian) group object as the lifting of the correspoding [[representable presheaf]] to a presheaf into (abelian) groups. This kind of lifting of some algebraic structure in sets to algebraic structure in a cartesian category makes sense when some category of algebras creates the limits needed to define them in sets.

\hypertarget{internalising_the_abelian_groups}{}\subsubsection*{{Internalising the abelian groups}}\label{internalising_the_abelian_groups}

If $M$ is a [[monoidal category]], then we can speak of [[monoid objects]] in $M$. However, we usually want $M$ to be somewhat like $Ab$ to think of monoid objects in $M$ as internal rings. For example, if $M$ is the category of abelian [[group objects]] in a cartesian monoidal category $C$, then we recreate the notion of ring object in $C$ from above. Or, if $M$ is any [[Ab-enriched category]], then it behaves enough like $Ab$ that we may consider its monoid objects as internal rings. There are yet other examples, however: a [[ring spectrum]] is a monoid object in [[spectra]], even though these are not $Ab$-enriched.

Other examples are [[simplicial ring]]s (as monoids in [[simplicial abelian group]]s) and [[dg-ring]]s, as well as the $A$-rings below.

\hypertarget{rings_over_a_ring_rings}{}\subsubsection*{{Rings over a ring ($A$-rings)}}\label{rings_over_a_ring_rings}

If $K$ is a commutative ring (or especially a [[field]]), then an [[associative algebra]] over $K$ is a monoid object in $K$-[[Mod]]; this is a special case of the previous section.

If $A$ is a noncommutative ring, then a \textbf{ring over $A$}, or simply an \textbf{$A$-ring}, is a monoid object $R$ in $A$-[[Bimod]] (that is, in $_A Mod _A$). Every $A$-ring is a ring in the usual sense, in the sense that there is an obvious [[forgetful functor]] to the usual rings. In fact the unit map $A \to R$ is a morphism of rings, and the category of $A$-rings is precisely the [[coslice category]] or under-category $A/Ring$. Thus by category-theoretic rules, one might be led to unconventionally call $A$-rings ``rings \emph{under} $A$''. Unfortunately, standard name for $A$-rings is ``rings \emph{over} $A$'', like conventionally calling $k$-algebras the ``algebras \emph{over} $K$''.

Unlike for the $k$-algebras, the multiplication $R\times R\to R$ which is the morphism of $A$-[[bimodules]], is not (left) $A$-linear in the \emph{second} factor, but only $A^{op}$-linear (that is, $A$-linear on the right). In other words, the axiom for $K$-algebras $k (r s) = r (k s)$ is not true, for $k\in A$, $r,s\in R$, although $k (r s) = (k r) s$ and $(r s) k = r (s k)$ do hold.

Both for a discussion for under-over and also for this difference between $K$-algebras and $A$-rings see the Caf\'e{}'s \href{http://golem.ph.utexas.edu/category/2008/12/a_quick_algebra_quiz.html}{quick algebra quiz}.

A dual notion to an $A$-ring is an $A$-[[coring]].

The structure of an $A\otimes A^{op}$-ring $(R,\mu,\eta)$ 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 $a,a'\in A$).

\hypertarget{higher_rings}{}\subsubsection*{{Higher rings}}\label{higher_rings}

By replacing in the sentence ``a ring is a [[monoid]] in [[Ab]]'' the [[abelian category]] [[Ab]] with a [[higher category theory|higher category]] of \emph{symmetric monoidal} higher groupoids, one obtains higher notion of rings.

Of particular interest is the maximal case of symmetric monoidal [[∞-groupoid]]s and, even more generally, that of [[spectrum|spectra]]. A [[monoid in an (∞,1)-category]] in the [[stable (∞,1)-category of spectra]] is an [[A-infinity-ring]] or [[associative ring spectrum]]. The commutative case is a [[commutative monoid in an (∞,1)-category]]: an [[E-infinity ring]] or [[commutative ring spectrum]].

\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\begin{example}
\label{}\hypertarget{}{}
\begin{itemize}%
\item The [[integers]] $\mathbb{Z}$ are a ring under the standard addition and multiplication operation.


\item For each $n$, this induces a ring structure on the [[cyclic group]] $\mathbb{Z}_n$, given by operations in $\mathbb{Z}$ modulo $n$ ([[cyclic rings]]).


\item The [[rational numbers]] $\mathbb{Q}$, [[real numbers]] $\mathbb{R}$ and [[complex numbers]] are rings under their standard operations, in fact these are even \emph{[[fields]]}.



\end{itemize}
\end{example}
\begin{example}
\label{}\hypertarget{}{}
For $R$ a ring, the [[polynomials]]

\begin{displaymath}
r_0 + r_1 x + r_2 x^2 + \cdots + r_n x^n
\end{displaymath}
(for arbitrary $n \in\mathbb{N}$) in a [[variable]] $x$ with [[coefficients]] in $R$ form another ring, the \emph{[[polynomial ring]]} denoted $R[x]$. This is the [[free construction|free]] $R$-[[associative algebra]] on a single generator $x$.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
For $R$ a ring and $n \in \mathbb{N}$, the set $M(n,R)$ of $n \times n$-[[matrices]] with [[coefficients]] in $R$ is a ring under elementwise addition and [[matrix multiplication]].

\end{example}
\begin{example}
\label{}\hypertarget{}{}
For $X$ a [[topological space]], the set of [[continuous functions]] $C(X,\mathbb{R})$ or $C(X,\mathbb{C})$ with values in the [[real numbers]] or [[complex numbers]] is a ring under pointwise (points in $X$) addition and multiplication.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
For $X$ a [[topological space]], the [[direct sum]] of its [[ordinary cohomology]] groups $H^\bullet(X,\mathbb{Z})$ forms a ring whose multiplication operation is the [[cup product]]. This is a [[graded object|graded ring]], graded by the cohomological degree.

\end{example}
\hypertarget{types_of_rings}{}\subsection*{{Types of rings}}\label{types_of_rings}

\begin{itemize}%
\item [[integral domain]]


\item [[principal ideal domain]]


\item [[Noetherian ring]]


\item many more\ldots{}



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[monoid]], [[monoid object]]


\item [[nonunital ring]], [[nonassociative ring]]


\item [[group]], [[group object]]


\item \textbf{ring}, [[E-∞ ring]]

\begin{itemize}%
\item [[associative algebra]]


\item [[commutative ring]], [[commutative algebra]]


\item [[ring object]]

\begin{itemize}%
\item [[topological ring]]


\item [[normed ring]]



\end{itemize}

\item [[localization of a ring]]


\item [[filtered ring]], [[associated graded ring]]


\item [[w-contractible ring]]


\item [[core of a ring]]



\end{itemize}

\item [[ideal]]


\item [[prime ring]]


\item [[spectrum of a commutative ring]]

\begin{itemize}%
\item [[affine scheme]], [[affine scheme]], [[spectral topological space]]

\end{itemize}

\item [[ring extension]],

\begin{itemize}%
\item [[infinitesimal extension]]

[[nilradical]], [[reduced ring]],



\end{itemize}

\item [[near-ring]]


\item [[module]], [[bimodule]]



\end{itemize}
\hypertarget{References}{}\subsection*{{References}}\label{References}

\hypertarget{ReferencesGeneral}{}\subsubsection*{{General}}\label{ReferencesGeneral}

Lecture notes include

\begin{itemize}%
\item Arno Fehm, \emph{Ringe} (\href{http://www.math.uni-konstanz.de/~fehm/teaching/algebra/geyer2.pdf}{pdf}) (in German)

\end{itemize}
\hypertarget{ReferencesHistory}{}\subsubsection*{{History}}\label{ReferencesHistory}

[[Richard Dedekind]] had introduced the concept today called \emph{ring} under the name \emph{Ordnung} (Ger: order, as in \href{http://en.wikipedia.org/wiki/Order_%28biology%29}{taxonomic order}). The word \emph{Zahlring} (Ger: number ring/ring of numbers) for this was introduced in section 9.31 of

\begin{itemize}%
\item [[David Hilbert]], \emph{Die Theorie der algebraischen Zahlk\"o{}rper}, Jahresbericht der Deutschen Mathematiker-Vereinigung 4 (1879)

\end{itemize}
There, the word ring just appears with a footnote mentioning Dedekind's use of the word ``Ordnung'', no further motivation is given. So probably Hilbert meant to use ``ring'' as in ``collection of things holding together'', not in the sense of circles or loops (as one might guess from the rings of [[cyclic groups]] $\mathbb{Z}_n$).

The first abstract axiomatic description of rings is in

\begin{itemize}%
\item Adolf Fraenkel, Journal f\"u{}r die reine und angewandte Mathematik \textbf{145} (1914)

\end{itemize}
which however contains some additional axioms not used anymore. The set of axioms in its modern form appears first in

\begin{itemize}%
\item [[Emmy Noether]], \emph{Ideal Theory in Rings}, Mathematische Annalen \textbf{83} (1921)

\end{itemize}
For historical accounts see

\begin{itemize}%
\item I. Kleiner, \emph{From numbers to rings: the early history of ring theory}, Elemente der Mathematik 53 (1998) 18-35. (\href{http://dx.doi.org/10.5169/seals-3627}{web})


\item \emph{The development of ring theory} (\href{http://www-history.mcs.st-and.ac.uk/HistTopics/Ring_theory.html}{pdf}) [[!redirects rings]]



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[[!redirects unital ring]] [[!redirects unital rings]] [[!redirects ring with unit]] [[!redirects rings with unit]] [[!redirects rings with units]] [[!redirects ring with identity]] [[!redirects rings with identity]] [[!redirects rings with identities]]

[[!redirects associative ring]] [[!redirects associative rings]]

[[!redirects associative unital ring]] [[!redirects associative unital rings]]

[[!redirects noncommutative ring]] [[!redirects noncommutative rings]]

[[!redirects non-commutative ring]] [[!redirects non-commutative rings]]

[[!redirects commutative ring]] [[!redirects commutative rings]]

[[!redirects commutative unital ring]] [[!redirects commutative unital rings]]



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