Cancellation Law for Ring Product of Integral Domain

Theorem
Let $\left({D, +, \circ}\right)$ be an integral domain whose zero is $0_D$.

Let $a \in D: a \ne 0_D$.

Then:


 * $\forall x, y \in D: a \circ x = a \circ y \implies x = y$

That is, all elements of $D^*$ are cancellable for ring product.

Proof
From the definition of integral domain, no elements of $D^*$ are zero divisors.

From Zero Divisor Not Cancellable, it follows that all elements of $D^*$ are cancellable for the ring product $\circ$.