Set of Positive Integers does not form Ring

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

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


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.