noncommutative algebra

Noncommutative algebra is a branch of mathematics that deals with algebraic structures where the order of operations matters. Unlike commutative algebra, where the product of two elements a and b is the same regardless of order (ab = ba), in noncommutative algebra, ab is not necessarily equal to ba.12

This concept is fundamental in modern algebra, which studies general algebraic structures like sets of numbers, matrices, and vector spaces. For instance, the set of n-by-n matrices forms a noncommutative ring, meaning matrix multiplication is not commutative. Quaternions, a number system invented by William Rowan Hamilton, are another example of a noncommutative structure. Noncommutative algebra has significant applications in various fields, including number theory and algebraic geometry.3