Exercise 1, § 3

For an integral ring $A$ to be factorial, it is necessary and sufficient that there exist a mapping $x \to s(x)$ from $A - \{0\}$ into $\mathbf{N}$, such that $s(xy) = s(x) + s(y)$, that the relation $s(x) = 0$ implies that x is invertible in A, and finally that for any two elements x, y of $A - \{0\}$, neither of which divides the other, there exist elements a, b, z, t of $A - \{0\}$ such that $ax + by = zt, \ s(z) < s(x), \ t$ being relatively prime to $x$ and to $y$. (If this condition is fulfilled, first show that every nonzero element of $A$ is a product of extremal elements, then that for every extremal element $p, Ap$ is a prime ideal; conclude with the aid of Theorem 1, $d$).