Category: Theorems in ring theory

Artin–Zorn theorem
In mathematics, the Artin–Zorn theorem, named after Emil Artin and Max Zorn, states that any finite alternative division ring is necessarily a finite field. It was first published in 1930 by Zorn, but
Posner's theorem
In algebra, Posner's theorem states that given a prime polynomial identity algebra A with center Z, the ring is a central simple algebra over , the field of fractions of Z. It is named after Ed Posner
Double centralizer theorem
In the branch of abstract algebra called ring theory, the double centralizer theorem can refer to any one of several similar results. These results concern the centralizer of a subring S of a ring R,
Freshman's dream
The freshman's dream is a name sometimes given to the erroneous equation , where is a real number (usually a positive integer greater than 1) and are nonzero real numbers. Beginning students commonly
Auslander–Buchsbaum theorem
In commutative algebra, the Auslander–Buchsbaum theorem states that regular local rings are unique factorization domains. The theorem was first proved by Maurice Auslander and David Buchsbaum. They sh
Auslander–Buchsbaum formula
In commutative algebra, the Auslander–Buchsbaum formula, introduced by Auslander and Buchsbaum , states that if R is a commutative Noetherian local ring and M is a non-zero finitely generated R-module
Primary decomposition
In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finit
Regev's theorem
In abstract algebra, Regev's theorem, proved by Amitai Regev , states that the tensor product of two PI algebras is a PI algebra.
Krull's separation lemma
In abstract algebra, Krull's separation lemma is a lemma in ring theory. It was proved by Wolfgang Krull in 1928.
Wedderburn–Artin theorem
In algebra, the Wedderburn–Artin theorem is a classification theorem for semisimple rings and semisimple algebras. The theorem states that an (Artinian) semisimple ring R is isomorphic to a product of
Eakin–Nagata theorem
In abstract algebra, the Eakin–Nagata theorem states: given commutative rings such that is finitely generated as a module over , if is a Noetherian ring, then is a Noetherian ring. (Note the converse
Theorem of transition
In algebra, the theorem of transition is said to hold between commutative rings if 1. * dominates ; i.e., for each proper ideal I of A, is proper and for each maximal ideal of B, is maximal 2. * for
Goldie's theorem
In mathematics, Goldie's theorem is a basic structural result in ring theory, proved by Alfred Goldie during the 1950s. What is now termed a right Goldie ring is a ring R that has finite uniform dimen
Cohen structure theorem
In mathematics, the Cohen structure theorem, introduced by Cohen, describes the structure of complete Noetherian local rings. Some consequences of Cohen's structure theorem include three conjectures o
Beauville–Laszlo theorem
In mathematics, the Beauville–Laszlo theorem is a result in commutative algebra and algebraic geometry that allows one to "glue" two sheaves over an infinitesimal neighborhood of a point on an algebra
Hilbert's syzygy theorem
In mathematics, Hilbert's syzygy theorem is one of the three fundamental theorems about polynomial rings over fields, first proved by David Hilbert in 1890, which were introduced for solving important
Hilbert's basis theorem
In mathematics, specifically commutative algebra, Hilbert's basis theorem says that a polynomial ring over a Noetherian ring is Noetherian.
Wedderburn's little theorem
In mathematics, Wedderburn's little theorem states that every finite domain is a field. In other words, for finite rings, there is no distinction between domains, division rings and fields. The Artin–
Gauss's lemma (polynomials)
In algebra, Gauss's lemma, named after Carl Friedrich Gauss, is a statement about polynomials over the integers, or, more generally, over a unique factorization domain (that is, a ring that has a uniq
Going up and going down
In commutative algebra, a branch of mathematics, going up and going down are terms which refer to certain properties of chains of prime ideals in integral extensions. The phrase going up refers to the
Jacobson density theorem
In mathematics, more specifically non-commutative ring theory, modern algebra, and module theory, the Jacobson density theorem is a theorem concerning simple modules over a ring R. The theorem can be
Mori–Nagata theorem
In algebra, the Mori–Nagata theorem introduced by Yoshiro Mori and Nagata, states the following: let A be a noetherian reduced commutative ring with the total ring of fractions K. Then the integral cl
Krull's principal ideal theorem
In commutative algebra, Krull's principal ideal theorem, named after Wolfgang Krull (1899–1971), gives a bound on the height of a principal ideal in a commutative Noetherian ring. The theorem is somet
Serre's criterion for normality
In algebra, Serre's criterion for normality, introduced by Jean-Pierre Serre, gives necessary and sufficient conditions for a commutative Noetherian ring A to be a normal ring. The criterion involves
Hopkins–Levitzki theorem
In the branch of abstract algebra called ring theory, the Akizuki–Hopkins–Levitzki theorem connects the descending chain condition and ascending chain condition in modules over semiprimary rings. A ri
Artin–Rees lemma
In mathematics, the Artin–Rees lemma is a basic result about modules over a Noetherian ring, along with results such as the Hilbert basis theorem. It was proved in the 1950s in independent works by th
Levitzky's theorem
In mathematics, more specifically ring theory and the theory of nil ideals, Levitzky's theorem, named after Jacob Levitzki, states that in a right Noetherian ring, every nil one-sided ideal is necessa
Multiplicity theory
In abstract algebra, multiplicity theory concerns the multiplicity of a module M at an ideal I (often a maximal ideal) The notion of the multiplicity of a module is a generalization of the degree of a
Kaplansky's theorem on projective modules
In abstract algebra, Kaplansky's theorem on projective modules, first proven by Irving Kaplansky, states that a projective module over a local ring is free; where a not-necessary-commutative ring is c
Nakayama's lemma
In mathematics, more specifically abstract algebra and commutative algebra, Nakayama's lemma — also known as the Krull–Azumaya theorem — governs the interaction between the Jacobson radical of a ring
Cartan–Brauer–Hua theorem
In abstract algebra, the Cartan–Brauer–Hua theorem (named after Richard Brauer, Élie Cartan, and Hua Luogeng) is a theorem pertaining to division rings. It says that given two division rings K ⊆ D suc
Skolem–Noether theorem
In ring theory, a branch of mathematics, the Skolem–Noether theorem characterizes the automorphisms of simple rings. It is a fundamental result in the theory of central simple algebras. The theorem wa