choosing generaters $f_1, \ldots , f_ t$ for $J$ we have

the Jacobian ideal of $B$ over $A$, namely the ideal in $B$ generated by the $r \times r$ minors of the matrx $(\partial f_ j/\partial x_ i)_{1 \leq i \leq r, 1 \leq j \leq t}$, contains the ideal $I^ cB$ for some $c$, and

the Cramer ideal of $B$ over $A$, namely the ideal in $B$ generated by the image in $B$ of the $r$th Fitting ideal of $J$ as an $A[x_1, \ldots , x_ r]^\wedge $-module, contains $I^ cB$ for some $c$.

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