Complex Numbers form Ring

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

Proof
From Complex Numbers under Addition form Abelian Group, $\struct {\C, +}$ 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:
 * $\forall x, y, z \in \C: x \times \paren {y \times z} = \paren {x \times y} \times z$

Thus $\struct{\C, +}$ is a semigroup.

Finally we have that Complex Multiplication Distributes over Addition:

Hence the result, by definition of ring.