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👉 In mathematics, a noncommutative ring is a ring in which the commutator of two elements (a
b) and (b
c) is not equal to the product of these two elements. The commutator of two elements a and b is defined as ab - ba. Noncommutative rings are important because they allow for more flexibility with operations, such as addition and multiplication, than do commutative rings. They also have applications in various fields such as algebra

noncommunally

👉 In mathematics, a noncommutative algebra is an algebraic structure that has both commutator and non-commutator operations. The non-commutativity of these operations means that the elements in the algebra do not commute with each other according to the usual rules of algebra. Noncommutative algebras are fundamental objects in many areas of mathematics, including group theory, representation theory, and quantum mechanics. They arise naturally in various contexts such as Lie groups, quantum groups, and quantum

noncommunal