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...

properties and polynomial identities. Commutative rings are much better understood than noncommutative ones. Algebraic geometry and algebraic number theory...

algebraic objects such as rings as well as geometric objects derived from them (e.g. by gluing along localizations or taking noncommutative stack quotients)....

area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described...

the ring of integers of a number field. Examples of noncommutative rings include the ring of n × n real square matrices with n ≥ 2, group rings in representation...

algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results...

Wedderburn's little theorem. The quaternions form a noncommutative domain. More generally, any division ring is a domain, since every nonzero element is invertible...

Abhandl.,3-14. Goodearl, An Introduction to Noncommutative Noetherian Rings Lam, First Course in Noncommutative Rings Obviously, multiplicatively closed sets...

division rings, and every non-field division ring is noncommutative. The best known example is the ring of quaternions. If one allows only rational instead...

the nil ideals are better understood than in noncommutative rings, primarily because in commutative rings, products involving nilpotent elements and sums...

"domain" for the general case including noncommutative rings. Some sources, notably Lang, use the term entire ring for integral domain. Some specific kinds...

polynomial ring in n variables with coefficients in the commutative ring R is the free commutative R-algebra of rank n, the noncommutative polynomial ring in...

Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces...

be applied to some noncommutative rings such as the set of Hurwitz quaternions. Let α and β represent two elements from such a ring. They have a common...

right-Noetherian. Noetherian rings are fundamental in both commutative and noncommutative ring theory since many rings that are encountered in mathematics...

of the ring. In general, if the nilradical is finitely generated (e.g., the ring is Noetherian), then it is nilpotent. For noncommutative rings, there...

mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if R and S are rings, then a ring homomorphism is...

matrices of a given dimension form a noncommutative ring, which is one of the most common examples of a noncommutative ring. If all entries of A below the main...

(2004), p. 244. Because simple commutative rings are fields. See Lam (2001). A First Course in Noncommutative Rings. p. 39. "Zero ideal". Math World. 22 Aug...

)\vert } ⁠. The smallest such ring R {\displaystyle R} has eight elements and it is the only noncommutative ring among 11 rings with unity of order 8, up...

"not necessarily associative", just as "noncommutative" means "not necessarily commutative" for noncommutative rings. An algebra is unital or unitary if it...

nonzero annihilator. The above definition applies also in the case of noncommutative rings, where the left annihilator of a left module is a left ideal, and...

3"); such operations are not commutative, and so are referred to as noncommutative operations. The idea that simple operations, such as the multiplication...

In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite...

the philosophy of noncommutative geometry, which tries to study various non-classical and/or pathological objects by noncommutative operator algebras...

In mathematics, a noncommutative unique factorization domain is a noncommutative ring with the unique factorization property. The ring of Hurwitz quaternions...

subject matter of noncommutative algebraic geometry and, more recently, of derived algebraic geometry. See also: Generic matrix ring. A homomorphism between...

spaces. If K is a noncommutative ring, then the concept still generalizes, with one caveat: since modules over noncommutative rings come in left and right...

abstract groups was O. K. Schmidt's 1916 Abstract Theory of Groups. Noncommutative ring theory began with extensions of the complex numbers to hypercomplex...

which is a simple noncommutative domain. These and many other nice examples are discussed in more detail in several noncommutative ring theory texts, including...