Ring theory
Content created by Egbert Rijke, Louis Wasserman, Fredrik Bakke, malarbol, Jonathan Prieto-Cubides, Maša Žaucer, Fernando Chu, Garrett Figueroa and Gregor Perčič.
Created on 2022-03-12.
Last modified on 2025-11-16.
Examples of rings
| Ring | File |
|---|---|
| Complex numbers | complex-numbers.large-ring-of-complex-numbers |
| Eisenstein integers | complex-numbers.eisenstein-integers |
| Endomorphism ring of an abelian group | group-theory.endomorphism-rings-abelian-groups |
| Gaussian integers | complex-numbers.gaussian-integers |
| Formal power series in commutative rings | commutative-algebra.formal-power-series-commutative-rings |
| Integers | elementary-number-theory.ring-of-integers |
| Integers modulo $𝑛$ | elementary-number-theory.standard-cyclic-rings |
| Rational numbers | elementary-number-theory.ring-of-rational-numbers |
| Real numbers | real-numbers.large-ring-of-real-numbers |
| Trivial ring | ring-theory.trivial-rings |
Modules in the ring theory namespace
module ring-theory where open import ring-theory.additive-orders-of-elements-rings public open import ring-theory.arithmetic-sequences-semirings public open import ring-theory.arithmetic-series-semirings public open import ring-theory.binomial-theorem-rings public open import ring-theory.binomial-theorem-semirings public open import ring-theory.category-of-cyclic-rings public open import ring-theory.category-of-rings public open import ring-theory.central-elements-rings public open import ring-theory.central-elements-semirings public open import ring-theory.characteristics-rings public open import ring-theory.commuting-elements-rings public open import ring-theory.congruence-relations-rings public open import ring-theory.congruence-relations-semirings public open import ring-theory.cyclic-rings public open import ring-theory.dependent-products-ring-extensions-rational-numbers public open import ring-theory.dependent-products-rings public open import ring-theory.dependent-products-semirings public open import ring-theory.division-rings public open import ring-theory.free-rings-with-one-generator public open import ring-theory.full-ideals-rings public open import ring-theory.function-rings public open import ring-theory.function-semirings public open import ring-theory.generating-elements-rings public open import ring-theory.geometric-sequences-rings public open import ring-theory.geometric-sequences-semirings public open import ring-theory.groups-of-units-rings public open import ring-theory.homomorphisms-cyclic-rings public open import ring-theory.homomorphisms-ring-extensions-rational-numbers public open import ring-theory.homomorphisms-rings public open import ring-theory.homomorphisms-semirings public open import ring-theory.ideals-generated-by-subsets-rings public open import ring-theory.ideals-rings public open import ring-theory.ideals-semirings public open import ring-theory.idempotent-elements-rings public open import ring-theory.initial-rings public open import ring-theory.integer-multiples-of-elements-rings public open import ring-theory.intersections-ideals-rings public open import ring-theory.intersections-ideals-semirings public open import ring-theory.invariant-basis-property-rings public open import ring-theory.invertible-elements-rings public open import ring-theory.isomorphisms-rings public open import ring-theory.joins-ideals-rings public open import ring-theory.joins-left-ideals-rings public open import ring-theory.joins-right-ideals-rings public open import ring-theory.kernels-of-ring-homomorphisms public open import ring-theory.large-function-rings public open import ring-theory.large-rings public open import ring-theory.left-ideals-generated-by-subsets-rings public open import ring-theory.left-ideals-rings public open import ring-theory.local-rings public open import ring-theory.localizations-rings public open import ring-theory.maximal-ideals-rings public open import ring-theory.multiples-of-elements-rings public open import ring-theory.multiples-of-elements-semirings public open import ring-theory.multiplicative-orders-of-units-rings public open import ring-theory.nil-ideals-rings public open import ring-theory.nilpotent-elements-rings public open import ring-theory.nilpotent-elements-semirings public open import ring-theory.nontrivial-rings public open import ring-theory.nonunital-left-algebras-rings public open import ring-theory.opposite-ring-extensions-rational-numbers public open import ring-theory.opposite-rings public open import ring-theory.partial-sums-sequences-semirings public open import ring-theory.poset-of-cyclic-rings public open import ring-theory.poset-of-ideals-rings public open import ring-theory.poset-of-left-ideals-rings public open import ring-theory.poset-of-right-ideals-rings public open import ring-theory.powers-of-elements-large-rings public open import ring-theory.powers-of-elements-rings public open import ring-theory.powers-of-elements-semirings public open import ring-theory.precategory-of-rings public open import ring-theory.precategory-of-semirings public open import ring-theory.products-ideals-rings public open import ring-theory.products-left-ideals-rings public open import ring-theory.products-right-ideals-rings public open import ring-theory.products-rings public open import ring-theory.products-subsets-rings public open import ring-theory.quotient-rings public open import ring-theory.radical-ideals-rings public open import ring-theory.right-ideals-generated-by-subsets-rings public open import ring-theory.right-ideals-rings public open import ring-theory.ring-extensions-rational-numbers public open import ring-theory.rings public open import ring-theory.semirings public open import ring-theory.subrings public open import ring-theory.subsets-rings public open import ring-theory.subsets-semirings public open import ring-theory.sums-of-finite-families-of-elements-rings public open import ring-theory.sums-of-finite-families-of-elements-semirings public open import ring-theory.sums-of-finite-sequences-of-elements-rings public open import ring-theory.sums-of-finite-sequences-of-elements-semirings public open import ring-theory.transporting-ring-structure-along-isomorphisms-abelian-groups public open import ring-theory.trivial-rings public
Recent changes
- 2025-11-16. Louis Wasserman. Large function commutative rings (#1719).
- 2025-11-13. Louis Wasserman. Standardize scalar multiplication on all sorts of rings (#1707).
- 2025-11-08. Louis Wasserman. Powers of real numbers (#1673).
- 2025-11-07. Louis Wasserman. Formula for sums of finite geometric series in the rational numbers (#1610).
- 2025-11-06. Louis Wasserman. The large ring of real numbers (#1664).