Ring is not Empty

Theorem
A ring cannot be empty.

Proof
In a ring $\struct {R, +, \circ}$, $\struct {R, +}$ forms a group.

From Group is not Empty, the group $\struct {R, +}$ contains at least the identity, so cannot be empty.

So every ring $\struct {R, +, \circ}$ contains at least the identity for ring addition.