Cancellation Law

In an algebraic structure $A$ with a binary operation $\cdot$, the left and right cancellation laws respectively hold if for all $x,y,z$ $$ x \cdot y = x \cdot z \Rightarrow y = z \,, $$ $$ x \cdot y = z \cdot y \Rightarrow x = z \ . $$ Such a structure is termed "cancellative".

A group satisfies the cancellation laws; a finite semi-group is a group if and only if it satisfies the cancellation laws. A commutative semi-group embeds in a group if and only if it is cancellative: for non-commutative groups the analogous statement does not hold in general, see Imbedding of semi-groups.

A ring is an integral domain if it is commutative and satisfies the cancellation laws for non-zero elements,