Simple question about relative primes in entire rings.

Question

Let $A$ be an entire ring. Let $a,b\in A$. Does

The g.c.d. of $a,b$ is a multiplicative unit. $\Rightarrow$ $\langle a,b\rangle=A$

hold? If yes, how can I proof it?

Answer 1

False, e.g. $\,\gcd(x,y)=1\,$ in $\,\Bbb Q[x,y]\,$ but $\,(x,y)\neq (1)$ else eval $\,xf\! +\!yg=1\,$ at $\,x,y=0\,$ $\,\Rightarrow 0 = 1.\ $