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...

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

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

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

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...

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

)\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...

coordinate ring of an affine algebraic variety, and the ring of integers of a number field. Examples of noncommutative rings include the ring of n × n real...

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

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...

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...

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...

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

noncommutative rings. Nil ideals are still associated with interesting open questions, especially the unsolved Köthe conjecture. In commutative rings...

maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields. In noncommutative ring theory, a maximal right...

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

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

of order p (cf. Prime field). A ring R is prime if and only if the zero ideal {0} is a prime ideal in the noncommutative sense. This being the case, the...

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

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...

ISBN 0-8218-0702-1, Zbl 0516.12001 *Lam, T. Y. (2001), A first course in noncommutative rings, Graduate Texts in Mathematics, vol. 131 (2nd ed.), New York: Springer-Verlag...

of Modern Science, 9 Smoktunowicz, Agata (2006), "Some results in noncommutative ring theory", International Congress of Mathematicians, Vol. II (PDF)...

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

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...

(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...

special name, such as normal subgroup for groups and two-sided ideals for rings. Kernels allow defining quotient objects (also called quotient algebras...

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...