Cancellation Law of Ring Product of Integral Domain

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Theorem

Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$.

Let $D^*$ denote $D \setminus \set {0_D}$, that is, $D$ without its zero.

Let $a \in D^*$.


Then:

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


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


Proof

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

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

$\blacksquare$


Sources