Cancellation Law of Multiplication

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 $\circ$.

$\blacksquare$

Sources

• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 1.4$: Theorem $2 \ \text{(vii)}$
• B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 1.3$: Lemma $1.4$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 55.4$