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.