Cancellation Law for Ring Product of Integral Domain

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

• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 4$. Elementary Properties: Theorem $2 \ \text{(vii)}$
• 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $3$: Some special classes of rings: Lemma $1.4$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 55.4$ Special types of ring and ring elements