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.

$\blacksquare$

Sources

• 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $2$: Some examples of rings: Some 'non-examples': $\text {(a)}$