Let $R$ be a commutative ring with identity having finite Krull dimension (denoted $\dim$). Let $Nil(R)$ be the set of all nilpotent elements in $R$, and let $J(R)$ the intersection of all maximal ideal of $R$.

If $Nil(R)\not= J (R)$, can we deduce than $\dim (R/J (R))<\dim (R)$? Or if the inequality is not true, under which conditions it can be true?