Proposition 3

Let $a$ and $b$ be two relatively prime ideals of a ring $A$. Let $a'$ and $b'$ be two ideals of $A$ such that every element of $a$ (resp. $b$) has a power in $a'$ (resp. $b'$). Then $a'$ and $b'$ are relatively prime.

Under the given hypothesis, every prime ideal which contains $a'$ contains $a$ and every prime ideal which contains $b'$ contains $b$. If a prime ideal contains $a'$ and $b'$, then it contains $a$ and $b$, which is absurd, since $a$ and $b$ are relatively prime; hence $a'$ and $b'$ are relatively prime.