Set of Positive Integers does not form Ring

Theorem
Let $\Z_{\ge 0}$ denote the set of positive integers.

Then the algebraic structure $\struct {\Z_{\ge 0}, +, \times}$ does not form a ring.

Proof
For $\struct {\Z_{\ge 0}, +, \times}$ to be a ring, it is necessary for the algebraic structure $\struct {\Z_{\ge 0}, +}$ to form a group.

But from the corollary to Natural Numbers under Addition do not form Group:
 * $\struct {\Z_{\ge 0}, +}$ is not a group.