Tour:Invertible implies cancellative in monoid

In a monoid (set with associative binary operation and neutral element) any invertible element can be canceled from an equation. In particular, in groups, any element is cancellative. The proof makes use of associativity in a crucial way.
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Statement

In a monoid (a set with an associative binary operation possessing a multiplicative identity element) the following are true:

• Any left invertible element (element having a left inverse) is left cancellative.
• Any right invertible element (element having a right inverse) is right cancellative.
• Any invertible element is cancellative.

Proof

We'll give here the proof for left invertible and left cancellative. An analogous proof works for right invertible and right cancellative.

Given: A monoid $M$ with binary operation $*$, and identity element (also called neutral element) $e$. $a\in M$ has a left inverse $b$ (i.e. an element $b*a=e$)

To prove: $a$ is left-cancellative: whenever $c,d\in M$ are such that $a*c=a*d$, then $c=d$

Proof: We start with:

$a*c=a*d$

Left-multiply both sides by $b$:

$b*(a*c)=b*(a*d)$

Use associativity:

$(b*a)*c=(b*a)*d$

We now use that $b*a=e$ is the identity element, to conclude that $c=d$.