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Inmathematics, anoncommutative ring is aring whose multiplication is notcommutative; that is, there exista andb in the ring such thatab andba are different. Equivalently, anoncommutative ring is a ring that is not acommutative ring.
Noncommutative algebra is the part ofring theory devoted to study of properties of the noncommutative rings, including the properties that apply also to commutative rings.
Sometimes the termnoncommutative ring is used instead ofring to refer to an unspecified ring which is not necessarily commutative, and hence may be commutative. Generally, this is for emphasizing that the studied properties are not restricted to commutative rings, as, in many contexts,ring is used as a shorthand forcommutative ring.
Although some authors do not assume that rings have a multiplicative identity, in this article we make that assumption unless stated otherwise.
Some examples of noncommutative rings:
Some examples of rings that are not typically commutative (but may be commutative in simple cases):
Beginning withdivision rings arising from geometry, the study of noncommutative rings has grown into a major area of modern algebra. The theory and exposition of noncommutative rings was expanded and refined in the 19th and 20th centuries by numerous authors. An incomplete list of such contributors includesE. Artin,Richard Brauer,P. M. Cohn,W. R. Hamilton,I. N. Herstein,N. Jacobson,K. Morita,E. Noether,Ø. Ore,J. Wedderburn and others.
Because noncommutative rings of scientific interest are more complicated than commutative rings, their structure, properties and behavior are less well understood. A great deal of work has been done successfully generalizing some results from commutative rings to noncommutative rings. A major difference between rings which are and are not commutative is the necessity to separately considerright ideals and left ideals. It is common for noncommutative ring theorists to enforce a condition on one of these types of ideals while not requiring it to hold for the opposite side. For commutative rings, the left–right distinction does not exist.
A division ring, also called a skew field, is aring in whichdivision is possible. Specifically, it is anonzero ring[2] in which every nonzero elementa has amultiplicative inverse, i.e., an elementx witha·x =x·a = 1. Stated differently, a ring is a division ring if and only if itsgroup of units is the set of all nonzero elements.
Division rings differ fromfields only in that their multiplication is not required to becommutative. However, byWedderburn's little theorem all finite division rings are commutative and thereforefinite fields. Historically, division rings were sometimes referred to as fields, while fields were called "commutative fields".
Amodule over a (not necessarily commutative) ring with unity is said to be semisimple (or completely reducible) if it is thedirect sum ofsimple (irreducible) submodules.
A ring is said to be (left)-semisimple if it is semisimple as a left module over itself. Surprisingly, a left-semisimple ring is also right-semisimple and vice versa. The left/right distinction is therefore unnecessary.
A semiprimitive ring or Jacobson semisimple ring or J-semisimple ring is a ring whoseJacobson radical is zero. This is a type of ring more general than asemisimple ring, but wheresimple modules still provide enough information about the ring. Rings such as the ring of integers are semiprimitive, and anartinian semiprimitive ring is just asemisimple ring. Semiprimitive rings can be understood assubdirect products ofprimitive rings, which are described by theJacobson density theorem.
A simple ring is a non-zeroring that has no two-sidedideal besides thezero ideal and itself. A simple ring can always be considered as asimple algebra. Rings which are simple as rings but not asmodules do exist: the fullmatrix ring over afield does not have any nontrivial ideals (since any ideal of M(n,R) is of the form M(n,I) withI an ideal ofR), but has nontrivial left ideals (namely, the sets of matrices which have some fixed zero columns).
According to theArtin–Wedderburn theorem, every simple ring that is left or rightArtinian is amatrix ring over adivision ring. In particular, the only simple rings that are a finite-dimensionalvector space over thereal numbers are rings of matrices over either the real numbers, thecomplex numbers, or thequaternions.
Any quotient of a ring by amaximal ideal is a simple ring. In particular, afield is a simple ring. A ringR is simple if and only if itsopposite ringRo is simple.
An example of a simple ring that is not a matrix ring over a division ring is theWeyl algebra.
Wedderburn's little theorem states that everyfinitedomain is afield. In other words, forfinite rings, there is no distinction between domains,division rings and fields.
TheArtin–Zorn theorem generalizes the theorem toalternative rings: every finite simple alternative ring is a field.[3]
The Artin–Wedderburn theorem is aclassification theorem forsemisimple rings andsemisimple algebras. The theorem states that an (Artinian)[4] semisimple ringR is isomorphic to aproduct of finitely manyni-by-nimatrix rings overdivision ringsDi, for some integersni, both of which are uniquely determined up to permutation of the indexi. In particular, anysimple left or rightArtinian ring is isomorphic to ann-by-nmatrix ring over adivision ringD, where bothn andD are uniquely determined.[5]
As a direct corollary, the Artin–Wedderburn theorem implies that every simple ring that is finite-dimensional over a division ring (a simple algebra) is amatrix ring. This isJoseph Wedderburn's original result.Emil Artin later generalized it to the case of Artinian rings.
TheJacobson density theorem is a theorem concerningsimple modules over a ringR.[6]
The theorem can be applied to show that anyprimitive ring can be viewed as a "dense" subring of the ring oflinear transformations of a vector space.[7][8] This theorem first appeared in the literature in 1945, in the famous paper "Structure Theory of Simple Rings Without Finiteness Assumptions" byNathan Jacobson.[9] This can be viewed as a kind of generalization of theArtin-Wedderburn theorem's conclusion about the structure ofsimpleArtinian rings.
More formally, the theorem can be stated as follows:
Let J(R) be theJacobson radical ofR. IfU is a right module over a ring,R, andI is a right ideal inR, then defineU·I to be the set of all (finite) sums of elements of the formu·i, where· is simply the action ofR onU. Necessarily,U·I is a submodule ofU.
IfV is amaximal submodule ofU, thenU/V issimple. SoU·J(R) is necessarily a subset ofV, by the definition of J(R) and the fact thatU/V is simple.[11] Thus, ifU contains at least one (proper) maximal submodule,U·J(R) is a proper submodule ofU. However, this need not hold for arbitrary modulesU overR, forU need not contain any maximal submodules.[12] Naturally, ifU is aNoetherian module, this holds. IfR is Noetherian, andU isfinitely generated, thenU is a Noetherian module overR, and the conclusion is satisfied.[13] Somewhat remarkable is that the weaker assumption, namely thatU is finitely generated as anR-module (and no finiteness assumption onR), is sufficient to guarantee the conclusion. This is essentially the statement of Nakayama's lemma.[14]
Precisely, one has the following.
A version of the lemma holds for right modules over non-commutativeunitary ringsR. The resulting theorem is sometimes known as theJacobson–Azumaya theorem.[15]
Localization is a systematic method of adding multiplicative inverses to aring, and is usually applied to commutative rings. Given a ringR and a subsetS, one wants to construct some ringR* andring homomorphism fromR toR*, such that the image ofS consists ofunits (invertible elements) inR*. Further one wantsR* to be the 'best possible' or 'most general' way to do this – in the usual fashion this should be expressed by auniversal property. The localization ofR byS is usually denoted byS −1R; however other notations are used in some important special cases. IfS is the set of the non zero elements of anintegral domain, then the localization is thefield of fractions and thus usually denoted Frac(R).
Localizingnon-commutative rings is more difficult; the localization does not exist for every setS of prospective units. One condition which ensures that the localization exists is theOre condition.
One case for non-commutative rings where localization has a clear interest is for rings of differential operators. It has the interpretation, for example, of adjoining a formal inverseD−1 for a differentiation operatorD. This is done in many contexts in methods fordifferential equations. There is now a large mathematical theory about it, namedmicrolocalization, connecting with numerous other branches. Themicro- tag is to do with connections withFourier theory, in particular.
Morita equivalence is a relationship defined betweenrings that preserves many ring-theoretic properties. It is named after Japanese mathematicianKiiti Morita who defined equivalence and a similar notion of duality in 1958.
Two ringsR andS (associative, with 1) are said to be (Morita)equivalent if there is an equivalence of the category of (left) modules overR,R-Mod, and the category of (left) modules overS,S-Mod. It can be shown that the left module categoriesR-Mod andS-Mod are equivalent if and only if the right module categoriesMod-R andMod-S are equivalent. Further it can be shown that any functor fromR-Mod toS-Mod that yields an equivalence is automaticallyadditive.
The Brauer group of afieldK is anabelian group whose elements areMorita equivalence classes ofcentral simple algebras of finite rank overK and addition is induced by thetensor product of algebras. It arose out of attempts to classifydivision algebras over a field and is named after the algebraistRichard Brauer. The group may also be defined in terms ofGalois cohomology. More generally, the Brauer group of ascheme is defined in terms ofAzumaya algebras.
The Ore condition is a condition introduced byØystein Ore, in connection with the question of extending beyondcommutative rings the construction of afield of fractions, or more generallylocalization of a ring. Theright Ore condition for amultiplicative subsetS of aringR is that fora ∈R ands ∈S, the intersectionaS ∩sR ≠ ∅.[16] A domain that satisfies the right Ore condition is called aright Ore domain. The left case is defined similarly.
Inmathematics,Goldie's theorem is a basic structural result inring theory, proved byAlfred Goldie during the 1950s. What is now termed a rightGoldie ring is aringR that has finiteuniform dimension (also called "finite rank") as a right module over itself, and satisfies theascending chain condition on rightannihilators of subsets ofR.
Goldie's theorem states that thesemiprime right Goldie rings are precisely those that have asemisimpleArtinian rightclassical ring of quotients. The structure of this ring of quotients is then completely determined by theArtin–Wedderburn theorem.
In particular, Goldie's theorem applies to semiprime rightNoetherian rings, since by definition right Noetherian rings have the ascending chain condition onall right ideals. This is sufficient to guarantee that a right-Noetherian ring is right Goldie. The converse does not hold: every rightOre domain is a right Goldie domain, and hence so is every commutativeintegral domain.
A consequence of Goldie's theorem, again due to Goldie, is that every semiprimeprincipal right ideal ring is isomorphic to a finite direct sum ofprime principal right ideal rings. Every prime principal right ideal ring is isomorphic to amatrix ring over a right Ore domain.
