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)....

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

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...

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

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...

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...

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

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

"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...

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...

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...

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

matrices of a given dimension form a noncommutative ring, which is one of the most common examples of a noncommutative ring. The determinant of a square matrix...

)\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...

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...

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

(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...

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...

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

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...

geometrical concepts. The study of rings that are not necessarily commutative is known as noncommutative algebra; it includes ring theory, representation theory...

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

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

pj (y) where i ≠ j, then xy = 0 in the product ring. Herstein, I.N. (2005) [1968], Noncommutative rings (5th ed.), Cambridge University Press, ISBN 978-0-88385-039-8...

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

A First Course in Noncommutative Rings, Berlin, New York: Springer-Verlag, ISBN 978-0-387-95325-0 Kelarev, Andrei V. (2002), Ring Constructions and Applications...

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