Extension Theorem for Distributive Operations/Identity

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 $*$.

Let $\circ'$ be the unique operation on $T$ which distributes over $*$ in $T$ and induces on $R$ the operation $\circ$.

Then:

If $e$ is an identity for $\circ$, then $e$ is also an identity for $\circ'$.

Proof

We have that $\circ'$ exists and is unique by Extension Theorem for Distributive Operations: Existence and Uniqueness.

Let $e$ be the identity element of $\struct {R, \circ}$.

Then the restrictions to $R$ of the endomorphisms $\lambda_e: x \mapsto e \circ' x$ and $\rho_e: x \mapsto x \circ' e$ of $\struct {T, *}$ are monomorphisms.

But then $\lambda_e$ and $\rho_e$ are monomorphisms by the Extension Theorem for Homomorphisms.

Hence it follows that $e$ is the identity element of $T$.

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$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $20$. The Integers: Theorem $20.8: 3^\circ$