(mathematics, physics) A version of C*-algebra which does away with commutativity, and the geometry which would be generated by/associated with such a modified algebra.

In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist a and b in the ring such that ab and ba are different.

In mathematics, noncommutative topology is a term used for the relationship between topological and C*-algebraic concepts.

a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as geometric objects derived from them (e.g. by gluing along localizations or taking noncommutative stack quotients).

In mathematical physics, noncommutative quantum field theory (or quantum field theory on noncommutative spacetime) is an application of noncommutative mathematics to the spacetime of quantum field theory that is an outgrowth of noncommutative geometry and index theory in which the coordinate functionsIt is possible to have a noncommuting time coordinate as in the paper by Doplicher, Fredenhagen and Roberts mentioned below, but this causes many problems such as the violation of unitarity of the S-matrix.