Let I be a proper ideal in R. Since R is a UFD, we can choose an element a from I having as few irreducible factors as possible. We claim that I = (a).

Of course, (a) is contained in I, so let b be an element of I. Then the ideal (a, b), which is contained in I, is principle by hypothesis. So there exists some element c (in I) such that (a, b) = (c). Consequently, there exist some elements r and s in R such that c = ra + sb. Since (a) is contained in (c), it follows that c divides a. So we write

c = c₁c₂ ^{. . .} c_m, and

a = c₁'c₂' ^{. . .} c_m'c_{m + 1}'c_{m + 2}' ^{. . .} c_n',

with m < n and such that each c_i and c_i' is irreducible, and c_i and c_i' are associates for each i < m.

On the other hand, since c is in I, it follows that n < m. (The integer n, which counts the number of irreducible factors of a, is the minimum number of irreducible factors that an element of I can have.) Therefore, n = m and so (a) = (c). But b is an element of (c) = (a). So I is contained in (a).

So (a) = I.