Ring theory

Ring theory explores algebraic structures called rings, where addition and multiplication behave much like they do with integers. The field investigates how rings are built, their representations (also known as modules), specific types such as group rings, division rings, and universal enveloping algebras, and related objects like rngs. It also covers properties like homological features and polynomial identities.

Commutative rings—where multiplication is commutative—are far better understood than noncommutative ones. Their development has been largely driven by algebraic geometry and algebraic number theory, which supply many natural examples. This area is now called commutative algebra, and it is so intertwined with those two fields that assigning a result to one over the others is often pointless. For instance, Hilbert's Nullstellensatz is fundamental to algebraic geometry but is stated and proved in commutative algebra. Similarly, Fermat's Last Theorem involves elementary arithmetic (part of commutative algebra), yet its proof relies on deep results from both algebraic number theory and algebraic geometry.

Commutative rings resemble familiar number systems, and many definitions aim to formalize properties of the integers. In this theory, ideals often replace numbers, with prime ideals capturing the essence of prime numbers. Integral domains—non-trivial commutative rings where no two non-zero elements multiply to zero—generalize another integer property and are the proper setting for studying divisibility. Principal ideal domains are integral domains where every ideal is generated by a single element, another integer trait. Euclidean domains are integral domains that support the Euclidean algorithm. Important examples come from polynomial rings and their factor rings. The hierarchy runs: Euclidean domain ⊂ principal ideal domain ⊂ unique factorization domain ⊂ integral domain ⊂ commutative ring.

Algebraic geometry mirrors commutative algebra. This correspondence began with Hilbert's Nullstellensatz, which links points of an algebraic variety to maximal ideals of its coordinate ring. This link was expanded to translate most geometric properties of varieties into algebraic properties of rings. Alexander Grothendieck later introduced schemes, a generalization of varieties built from any commutative ring. Specifically, the spectrum of a commutative ring is the space of its prime ideals with the Zariski topology, augmented with a sheaf of rings. These affine schemes generalize affine varieties, and general schemes are formed by gluing together such affine schemes, much like constructing a manifold from atlas charts.

Noncommutative rings differ in flavor, allowing more unusual behavior. They often resemble rings of matrices. Following the algebraic geometry model, recent efforts have aimed to define noncommutative geometry based on noncommutative rings, treating them as rings of functions on hypothetical "noncommutative spaces." This trend began in the 1980s with noncommutative geometry and the discovery of quantum groups, leading to a better understanding of noncommutative rings, especially noncommutative Noetherian ones.

Noncommutative rings and associative algebras (rings that are also vector spaces) are often studied through their module categories. A module over a ring is an abelian group acted on by the ring as a ring of endomorphisms, similar to how fields (integral domains where every non-zero element is invertible) act on vector spaces. Examples include rings of square matrices, rings of endomorphisms of abelian groups or modules, and monoid rings.

Representation theory draws heavily on noncommutative rings. It studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and examines modules over these structures. This makes an abstract object more concrete by describing its elements with matrices and algebraic operations via matrix addition and multiplication (which is noncommutative). Groups, associative algebras, and Lie algebras are amenable to this approach, with group representation theory being the most prominent and historically first.