Definition:Additive Group of Integers

Theorem
The set of integers under addition $$\left({\mathbb{Z}, +}\right)$$forms an abelian group.

Thus it follows that integer addition is:


 * well-defined on $$\mathbb{Z}$$;
 * closed on $$\mathbb{Z}$$;
 * associative on $$\mathbb{Z}$$;
 * commutative on $$\mathbb{Z}$$.

Proof
From the definition of the integers, the algebraic structure $$\left({\mathbb{Z}, +}\right)$$ is an isomorphic copy of the inverse completion of $$\left ({\mathbb{N}, +}\right)$$.

As the Natural Numbers are a Naturally Ordered Semigroup, it follows that:
 * $$\left ({\mathbb{N}, +}\right)$$ is a commutative semigroup;
 * all elements of $$\left ({\mathbb{N}, +}\right)$$ are cancellable.

The result follows from Inverse Completion a Group.