symmetric monoidal (∞,1)-category of spectra
A ring (also: number ring) is a basic structure in algebra: a set equipped with two binary operations called addition and multiplication, such that the operation of addition forms an abelian group and the operation of multiplication a monoid structure which 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; the polynomial rings, etc.
More abstractly, a ring is a monoid internal to abelian groups (with their tensor product of abelian groups), and this prespective helps to explain the central relevance of the concept, owing to the fundamental nature of the notion of monoid objects. Accordingly monoids internal to other abelian categories and more generally stable infinity-categories constitute generalizations of the notion of 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.
A ring (unital and not-necessarily commutative) is an abelian group $R$ equipped with
an element $1 \in R$
a bilinear map, hence a group homomorphism
out of the tensor product of abelian groups,
such that $\cdot$ is associative and unital with respect to 1.
The fact that the product is a bilinear map is the distributivity law: for all $r, r_1, r_2 \in R$ we have
and
A (unital, non-commutative) ring is (equivalently)
A commutative (unital) ring is an commutative monoid object in $(Ab, \otimes)$.
In usual ring theory people often talk about 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 commutative nonunital. Note the adjective ‘nonunital’ is an example of the red herring principle.
It is possible to internalise the notion of ring in at least two different ways. Either one can replace the 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 $M$.
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 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.
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 rings (as monoids in simplicial abelian groups) and dg-rings, as well as the $A$-rings below.
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 ring over $A$, or simply an $A$-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 under $A$”. Unfortunately, standard name for $A$-rings is “rings over $A$”, like conventionally calling $k$-algebras the “algebras 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 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 Café's 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 $a,a'\in A$).
By replacing in the sentence “a ring is a monoid in Ab” the abelian category Ab with a higher category of symmetric monoidal higher groupoids, one obtains higher notion of rings.
Of particular interest is the maximal case of symmetric monoidal ∞-groupoids and, even more generally, that of 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.
The integers $\mathbb{Z}$ are a ring under the standard addition and multiplication operation.
For each $n$, this induces a ring structure on the cyclic group $\mathbb{Z}_n$, given by operations in $\mathbb{Z}$ modulo $n$.
The rational numbers $\mathbb{Q}$, real numbers $\mathbb{R}$ and complex numbers are rings under their standard operations, in fact these are even fields.
For $R$ a ring, the polynomials
(for arbitrary $n \in\mathbb{N}$) in a variable $x$ with coefficients in $R$ form another ring, the polynomial ring denoted $R[x]$. This is the free $R$-associative algebra on a single generator $x$.
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.
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.
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 ring, graded by the cohomological degree.
ring, E-∞ ring
Lecture notes include
Richard Dedekind had introduced the concept today called ring under the name Ordnung (Ger: order, as in taxonomic order). The word Zahlring (Ger: number ring/ring of numbers) for this was introduced in section 9.31 of
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
which however contains some additional axioms not used anymore. The set of axioms in its modern form appears first in
For historical accounts see
I. Kleiner, From numbers to rings: the early history of ring theory, Elemente der Mathematik 53 (1998) 18-35. (web)
The development of ring theory (pdf)