Complex Numbers form Ring

Theorem
The set of complex numbers $\C$ forms a ring under addition and multiplication: $\left({\C, +, \times}\right)$.

Proof
From Complex Numbers under Addition form Abelian Group, $\left({\C, +}\right)$ is an abelian group.

We also have that:
 * Complex Multiplication is Closed:
 * $\forall x, y \in \C: x \times y \in \C$


 * Complex Multiplication is Associative:
 * $\forall x, y, z \in \C: x \times \left({y \times z}\right) = \left({x \times y}\right) \times z$

Thus $\left({\C, +}\right)$ is a semigroup.

Finally we have that Complex Multiplication Distributes over Addition:
 * $\forall x, y, z \in \C:$
 * $x \times \left({y + z}\right) = x \times y + x \times z$
 * $\left({y + z}\right) \times x = y \times x + z \times x$

Hence the result, by definition of ring.