Coprime elements generate coprime ideals

Question

Is this true in any commutative rings? I.e. $$\gcd(a,b)=1\implies (a)+(b)=R$$ I think there must be some conditions on the ring to make this implication otherwise it does not work.

This may related to this question here.

Answer 1

This is not true in arbitrary rings. For (counter)example, in a polynomial ring in two variables $R = k[x,y]$, we have $\gcd(x,y)=1$, but $(x)+(y) \neq R$.

It is true in Bézout domains. I don’t know if it’s equivalent to being a Bézout domain.