noncommutative algebra

In mathematics, more specifically modern algebra and ring theory, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exists a and b in R with a·'b ≠ b·a. Note that many authors use the term noncommutative ring to refer to rings which are not necessarily commutative, and hence include commutative rings in their definition. Noncommutative algebra is the study of results applying to rings that are not required to be commutative; however, many important results in this area apply to commutative rings as special cases.