Greatest common divisor theorem proof

Question

I wanted to prove an algebraic theorem and therefore I would need a statement like the following:

In a commutative Ring $R$ it states for $a,b,c \in R$ with $a \neq b, a \leq c, b \leq c$ that $$\text{gcd}(a,c)=1 \wedge \text{gcd}(b,c)=1 \Longrightarrow \text{gcd}(a,b)=1.$$

I don't know if this is right in general. I couldn't find a counterexample so my hope is that it's true.

Is there a general proof?

Answer 1

Take the ring $\mathbb{Z}$. Take $c = 7, \; a = 2, \; b = 4$

Then $gcd(2,7) = 1, \; gcd(4, 7) = 1$, but $gcd(2,4) = 2$. In general this is false.