For a commutative ring $R $ with identity, if $Nil (R)\not= J (R)$, can we deduce than $dim (R/J (R))<dim (R)$? ($R $ has finite $Krull$ dimension (=$dim$), $Nil (R)$=the set of all nilpotent elements, $J (R)$=the intersection of all maximal ideal of $R $) (Or if the inequality is not true, under which conditions it can be true)

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?

For a commutative ring $R $ with identity, if $Nil (R)\not= J (R)$, can we deduce than $dim (R/J (R))<dim (R)$? ($R $ has finite $Krull$ dimension (=$dim$), $Nil (R)$=the set of all nilpotent elements, $J (R)$=the intersection of all maximal ideal of $R $)

For a commutative ring $R $ with identity, if $Nil (R)\not= J (R)$, can we deduce than $dim (R/J (R))<dim (R)$? ($R $ has finite $Krull$ dimension (=$dim$), $Nil (R)$=the set of all nilpotent elements, $J (R)$=the intersection of all maximal ideal of $R $) (Or if the inequality is not true, under which conditions it can be true)