Noncommutative ring

In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist a and b in the ring such that ab and ba are different. Equivalently, a noncommutative ring is a ring that is not a commutative ring.

Noncommutative algebra is the part of ring theory devoted to study of properties of the noncommutative rings, including the properties that apply also to commutative rings.

Sometimes the term noncommutative ring is used instead of ring 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 for commutative 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:

• The matrix ring of n-by-n matrices over the real numbers, where n > 1
• Hamilton's quaternions
• Any group ring constructed from a group that is not abelian

Some examples of rings that are not typically commutative (but may be commutative in simple cases):

• The
  free ring
  ${\displaystyle \mathbb {Z} \langle x_{1},\ldots ,x_{n}\rangle }$ generated by a finite set, an example of two non-equal elements being ${\displaystyle 2x_{1}x_{2}+x_{2}x_{1}\neq 3x_{1}x_{2}}$
• The Weyl algebra ${\displaystyle A_{n}(\mathbb {C} )}$, being the ring of polynomial differential operators defined over affine space; for example, ${\displaystyle A_{1}(\mathbb {C} )\cong \mathbb {C} \langle x,y\rangle /(xy-yx-1)}$, where the ideal corresponds to the commutator
• The quotient ring ${\displaystyle \mathbb {C} \langle x_{1},\ldots ,x_{n}\rangle /(x_{i}x_{j}-q_{ij}x_{j}x_{i})}$, called a quantum plane, where ${\displaystyle q_{ij}\in \mathbb {C} }$
• Any Clifford algebra can be described explicitly using an algebra presentation: given an ${\displaystyle \mathbb {F} }$-vector space ${\displaystyle V}$ of dimension n with a quadratic form ${\displaystyle q:V\otimes V\to \mathbb {F} }$, the associated Clifford algebra has the presentation ${\displaystyle \mathbb {F} \langle e_{1},\ldots ,e_{n}\rangle /(e_{i}e_{j}+e_{j}e_{i}-q(e_{i},e_{j}))}$ for any basis ${\displaystyle e_{1},\ldots ,e_{n}}$ of ${\displaystyle V}$,
• Superalgebras are another example of noncommutative rings; they can be presented as ${\displaystyle \mathbb {C} [x_{1},\ldots ,x_{n}]\langle \theta _{1},\ldots ,\theta _{m}\rangle /(\theta _{i}\theta _{j}+\theta _{j}\theta _{i})}$
• There are finite noncommutative rings: for example, the n-by-n matrices over a
  isomorphic to it or to its opposite.^[1]

A division ring, also called a skew field, is a

Division rings differ from

A module over a (not necessarily commutative) ring with unity is said to be semisimple (or completely reducible) if it is the direct sum of simple (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 whose

A simple ring is a non-zero

Any quotient of a ring by a maximal ideal is a simple ring. In particular, a field is a simple ring. A ring R is simple if and only if its opposite ring R^o is simple.

An example of a simple ring that is not a matrix ring over a division ring is the Weyl algebra.

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

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 a matrix ring. This is Joseph Wedderburn's original result. Emil Artin later generalized it to the case of Artinian rings.

The Jacobson density theorem is a theorem concerning simple modules over a ring R.^[6]

The theorem can be applied to show that any

The Jacobson Density Theorem. Let U be a simple right R-module, D = End(U_R), and X ⊂ U a finite and D-linearly independent set. If A is a D-linear transformation on U then there exists r ∈ R such that A(x) = x · r for all x in X.[10]

Let J(R) be the Jacobson radical of R. If U is a right module over a ring, R, and I is a right ideal in R, then define U·I to be the set of all (finite) sums of elements of the form u·i, where · is simply the action of R on U. Necessarily, U·I is a submodule of U.

If V is a

Nakayama's lemma: Let U be a finitely generated right module over a ring R. If U is a non-zero module, then U·J(R) is a proper submodule of U.^[14]

A version of the lemma holds for right modules over non-commutative

Localization is a systematic method of adding multiplicative inverses to a ring, and is usually applied to commutative rings. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units (invertible elements) in R*. Further one wants R* to be the 'best possible' or 'most general' way to do this – in the usual fashion this should be expressed by a universal property. The localization of R by S is usually denoted by S ⁻¹R; however other notations are used in some important special cases. If S is the set of the non zero elements of an integral domain, then the localization is the field of fractions and thus usually denoted Frac(R).

Localizing

Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. It is named after Japanese mathematician Kiiti Morita who defined equivalence and a similar notion of duality in 1958.

Two rings R and S (associative, with 1) are said to be (Morita) equivalent if there is an equivalence of the category of (left) modules over R, R-Mod, and the category of (left) modules over S, S-Mod. It can be shown that the left module categories R-Mod and S-Mod are equivalent if and only if the right module categories Mod-R and Mod-S are equivalent. Further it can be shown that any functor from R-Mod to S-Mod that yields an equivalence is automatically

The Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras of finite rank over K and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is named after the algebraist Richard Brauer. The group may also be defined in terms of Galois cohomology. More generally, the Brauer group of a scheme is defined in terms of Azumaya algebras.

The Ore condition is a condition introduced by

In

• Derived algebraic geometry
• Noncommutative geometry
• Noncommutative algebraic geometry
• Noncommutative harmonic analysis
• Representation theory (group theory)