Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




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 n\mathbb{Z}_n under their mod-nn operations inherited from the integers (cyclic rings); the polynomial rings, etc.

More abstractly, a ring is a monoid 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 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 RR equipped with

  1. an element 1R1 \in R

  2. a bilinear map, hence a group homomorphism

    :RRR \cdot : R \otimes R \to R

    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 2Rr, r_1, r_2 \in R we have

r(r 1+r 2)=rr 1+rr 2 r \cdot (r_1 + r_2) = r \cdot r_1 + r \cdot r_2


(r 1+r 2)r=r 1r+r 2r. (r_1 + r_2) \cdot r = r_1 \cdot r + r_2 \cdot r \,.

A (unital, non-commutative) ring is (equivalently)

A commutative (unital) ring is a commutative monoid object in (Ab,)(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 AbAb. 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.


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)(1 + a)(1 + b) in two different ways, and cancel to conclude a+b=b+aa + b = b + a.) The result is false for nonunital rings: for any group (G,+)(G, +) we could define multiplication to be the constant function at the additive identity, and all the axioms except additive commutativity are trivially satisfied.


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 CC – see ring object – , or one can replace Ab in the fancy definition with another category MM.

Internalising the sets

If CC is a cartesian monoidal category, then any Lawvere theory may be internalised in CC. The theory of rings is an example, so we can speak of ring objects in CC. Then a ring object in SetSet 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 RR in a topos 𝒯\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.

Internalising the abelian groups

If MM is a monoidal category, then we can speak of monoid objects in MM. However, we usually want MM to be somewhat like AbAb to think of monoid objects in MM as internal rings. For example, if MM is the category of abelian group objects in a cartesian monoidal category CC, then we recreate the notion of ring object in CC from above. Or, if MM is any Ab-enriched category, then it behaves enough like AbAb 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 AbAb-enriched.

Other examples are simplicial rings (as monoids in simplicial abelian groups) and dg-rings, as well as the AA-rings below.

Rings over a ring (AA-rings)

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

If AA is a noncommutative ring, then a ring over AA, or simply an AA-ring, is a monoid object RR in AA-Bimod (that is, in AMod A_A Mod _A). Every AA-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 ARA \to R is a morphism of rings, and the category of AA-rings is precisely the coslice category or under-category A/RingA/Ring. Thus by category-theoretic rules, one might be led to unconventionally call AA-rings “rings under AA”. Unfortunately, standard name for AA-rings is “rings over AA”, like conventionally calling kk-algebras the “algebras over KK”.

Unlike for the kk-algebras, the multiplication R×RRR\times R\to R which is the morphism of AA-bimodules, is not (left) AA-linear in the second factor, but only A opA^{op}-linear (that is, AA-linear on the right). In other words, the axiom for KK-algebras k(rs)=r(ks)k (r s) = r (k s) is not true, for kAk\in A, r,sRr,s\in R, although k(rs)=(kr)sk (r s) = (k r) s and (rs)k=r(sk)(r s) k = r (s k) do hold.

Both for a discussion for under-over and also for this difference between KK-algebras and AA-rings see the Café's quick algebra quiz.

A dual notion to an AA-ring is an AA-coring.

The structure of an AA opA\otimes A^{op}-ring (R,μ,η)(R,\mu,\eta) is determined by the structure of AA as a ring, together with the two natural homomorphisms of rings s=η(1 A):ARs = \eta(-\otimes 1_A):A\to R and t=η(1 A):A opRt=\eta(1_A\otimes -):A^{op}\to R which have commuting images (s(a)t(a)=t(a)s(a)s(a)t(a')=t(a')s(a), for all a,aAa,a'\in A).

Higher rings

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.



For RR a ring, the polynomials

r 0+r 1x+r 2x 2++r nx n r_0 + r_1 x + r_2 x^2 + \cdots + r_n x^n

(for arbitrary nn \in\mathbb{N}) in a variable xx with coefficients in RR form another ring, the polynomial ring denoted R[x]R[x]. This is the free RR-associative algebra on a single generator xx.


For RR a ring and nn \in \mathbb{N}, the set M(n,R)M(n,R) of n×nn \times n-matrices with coefficients in RR is a ring under elementwise addition and matrix multiplication.


For XX a topological space, the set of continuous functions C(X,)C(X,\mathbb{R}) or C(X,)C(X,\mathbb{C}) with values in the real numbers or complex numbers is a ring under pointwise (points in XX) addition and multiplication.


For XX a topological space, the direct sum of its ordinary cohomology groups H (X,)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.

Types of rings



Lecture notes include

  • Arno Fehm, Ringe (pdf) (in German)


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

  • David Hilbert, Die Theorie der algebraischen Zahlkörper, Jahresbericht der Deutschen Mathematiker-Vereinigung 4 (1879)

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 n\mathbb{Z}_n).

The first abstract axiomatic description of rings is in

  • Adolf Fraenkel, Journal für die reine und angewandte Mathematik 145 (1914)

which however contains some additional axioms not used anymore. The set of axioms in its modern form appears first in

  • Emmy Noether, Ideal Theory in Rings, Mathematische Annalen 83 (1921)

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)

Revised on September 29, 2017 05:03:27 by Urs Schreiber (