A noncommutative ring 
is a ring in which the law of multiplicative
commutativity is not satisfied, i.e.,
 |
for any two elements 
. In such a case, the elements 
and 
of the ring 
are said not to commute. An important example of a noncommutative
ring is the ring 
consisting of all 
matrices whose elements are members of the field 
.
