Coprime elements generate coprime ideals 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.

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.
