Why is $I + (a) = R$?

Question

Let $I$ be a maximal ideal of a ring $R$ and let $(a)$ be the principal ideal generated by the element $a$ which lies in $R$ but not $I$.
Why does $R = I + (a)$?

Thanks in advance

Answer 1

$I$ is a maximal ideal means if $J$ is any ideal of $R$ with $I \subseteq J$ then either $I = J$ or $J = R.$ In this case, take $J = I + (a).$ By the given condition, $J \neq I.$ So $J = R.$