Let $R$ be a commutative ring with identity. Let $a,b$ be elements of $R$. If $b+(a)$ is not a zero divisor in $R/(a)$, does it follow that $(a,b)=R$ ?
The converse can be easily shown to be true. More explicitly, if $(a,b)=R$, then $b+(a)$ is not a zero divisor in $R/(a)$.
Thank you
Consider $R=S[a,b]$ for any commutative ring $S$ and indeterminates $a$ and $b$.
Alternatively, notice $ax+by=1$ in $R$ implies $b$ is invertible in $R/(a)$, and invertibility doesn't follow from simply not being a zero divisor (as the $b\in S[b]$ example is an illustration of).