# Extension Theorem for Distributive Operations

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## 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 the following hold:

### Existence and Uniqueness

- There exists a unique operation $\circ'$ on $T$ which distributes over $*$ in $T$ and induces on $R$ the operation $\circ$.

### Associativity

- If $\circ$ is associative, then so is $\circ'$.

### Commutativity

- If $\circ$ is commutative, then so is $\circ'$

### Identity

### Cancellability

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

## Sources

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