View source for Cancellation law

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