Ring (mathematics)

Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers.

The conceptualization of rings spanned the 1870s to the 1920s, with key contributions by Dedekind, Hilbert, Fraenkel, and Noether.

A ring is a set R equipped with two binary operations[a] + (addition) and ⋅ (multiplication) satisfying the following three sets of axioms, called the ring axioms:[1][2][3] In notation, the multiplication symbol · is often omitted, in which case a · b is written as ab.

As explained in § History below, many authors apply the term "ring" without requiring a multiplicative identity.

⁠ consisting of the numbers The axioms of a ring were elaborated as a generalization of familiar properties of addition and multiplication of integers.

[13] 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),[citation needed] 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).

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.

A subset S of R is called a subring if any one of the following equivalent conditions holds: For example, the ring ⁠

For commutative rings, the ideals generalize the classical notion of divisibility and decomposition of an integer into prime numbers in algebra.

Clearly the direct sum of such ideals also defines a product of rings that is isomorphic to R. Equivalently, the above can be done through central idempotents.

[37] For example, choosing a basis, a symmetric algebra satisfies the universal property and so is a polynomial ring.

Given a non-constant monic polynomial f in R[t], there exists a ring S containing R such that f is a product of linear factors in S[t].

The Hilbert's Nullstellensatz (theorem of zeros) states that there is a natural one-to-one correspondence between the set of all prime ideals in

In particular, many local problems in algebraic geometry may be attacked through the study of the generators of an ideal in a polynomial ring.

The set of all square matrices of size n with entries in R forms a ring with the entry-wise addition and the usual matrix multiplication.

consisting of (xn) such that xj maps to xi under Rj → Ri, j ≥ i.

The localization generalizes the construction of the field of fractions of an integral domain to an arbitrary ring and modules.

The field of fractions of an integral domain R is the localization of R at the prime ideal zero.

⁠ at the principal ideal (p) generated by a prime number p; it is called the ring of p-adic integers and is denoted ⁠

On the other hand, the interaction between the integral closure and completion has been among the most important aspects that distinguish modern commutative ring theory from the classical one developed by the likes of Noether.

[47] Let V be a finite-dimensional vector space over a field k and f : V → V a linear map with minimal polynomial q.

The study of conjugacy classes figures prominently in the classical theory of division rings; see, for example, the Cartan–Brauer–Hua theorem.

is a matrix ring over F (that is, A is split by F.) If the extension is finite and Galois, then Br(F / k) is canonically isomorphic to

For example: Many different kinds of mathematical objects can be fruitfully analyzed in terms of some associated ring.

The ring structure in cohomology provides the foundation for characteristic classes of fiber bundles, intersection theory on manifolds and algebraic varieties, Schubert calculus and much more.

This ring reflects many of the combinatorial properties of the simplicial complex, so it is of particular interest in algebraic combinatorics.

In particular, the algebraic geometry of the Stanley–Reisner ring was used to characterize the numbers of faces in each dimension of simplicial polytopes.

Consider those endomorphisms of A, that "factor through" right (or left) multiplication of R. In other words, let EndR(A) be the set of all morphisms m of A, having the property that m(r ⋅ x) = r ⋅ m(x).

and a unit map S → X from the sphere spectrum S, such that the ring axiom diagrams commute up to homotopy.