# 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.

$\struct {\Z_{\ge 0}, +}$ is not a group.

$\blacksquare$