Extension Theorem for Distributive Operations/Cancellability

Theorem

Let $\struct {R, *}$ be a commutative semigroup, all of whose elements are cancellable.

Let $\struct {T, *}$ be an inverse completion of $\struct {R, *}$.

Let $\circ$ be an operation on $R$ which distributes over $*$.

Then:

Every element of $R$ cancellable for $\circ$ is also cancellable for $\circ'$.

Proof

Let $a$ be an element of $R$ cancellable for $\circ$.

Then the restrictions to $R$ of the endomorphisms:

$\lambda_a: x \mapsto a \circ' x$

$\rho_a: x \mapsto x \circ' a$

of $\struct {T, *}$ are monomorphisms.

But then $\lambda_a$ and $\rho_a$ are monomorphisms by the Extension Theorem for Homomorphisms.

Hence $a$ is cancellable for $\circ'$.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $20$. The Integers: Theorem $20.8: 4^\circ$