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A **noncommutative ring** is a type of algebraic structure that generalizes some properties of familiar number systems, like the integers or polynomials, but allows for multiplication where the order of the factors matters. In other words, in a noncommutative ring, it is possible for the product of two elements \( a \) and \( b \) to differ from the product \( b \) and \( a \); that is, \( ab \neq ba \).