cancelability

Cancelability is a mathematical property describing when a common factor can be removed from both sides of an equation without changing the solutions. In algebraic structures such as semigroups or monoids, an element a is left-cancelable if ab = ac implies b = c for every b, c; it is right-cancelable if ba = ca implies b = c. An element that is both left- and right-cancelable is simply cancelable. In a group, where every element has an inverse, all nonzero elements are cancelable, and ab = ac implies b = c (via multiplying on the left by a^{-1}).

In rings or modules, cancellation can depend on the properties of the element. If a is not a zero divisor in a ring, ab = ac implies b = c. If a is invertible, cancellation holds for both sides. In matrix algebra, a is left (or right) cancellable precisely when it is invertible; otherwise, cancellation can fail as noninvertible matrices may satisfy Am = An with m ≠ n.

Non-cancellative structures exist as well. For example, in any structure with a zero element, 0·b = 0·c for all b, c, so left cancellation fails for a = 0. Likewise, in rings with zero divisors or in certain modular arithmetic settings, cancellation can fail when factors are zero divisors.

Applications include solving equations, simplifying fractions or rational expressions, and reducing algebraic expressions. Related ideas appear in category theory (left- and right-cancelable morphisms) and in logic or computer algebra, where cancellation rules are used with care to avoid dividing by zero or introducing extraneous solutions.