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...

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)....

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...

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...

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

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...