Let $A \colon= {\Bbb F}_p[[X_1,\ldots,X_n]]$ be a $n$-variable power series ring over a finite field ${\Bbb F}_p$. We put ${\frak m}_A \colon= (X_1,\ldots,X_n)$.
Definition(Super-Gorenstein ideal): ${\frak a} = (F_1,...,F_d)$ is called a ‘‘super-Gorenstein ideal” of $A$, if for each $k \geq 1$ there exists an ideal ${\frak b}_k \in {\frak m}_A^k$ such that
${\frak b}_k$ has height $n$ and is generated as ${\frak b}_k = (f_{k,1},...,f_{k,n})$ with each $f_{k,i} \in {\frak m}_A^k$.
${\frak a} + {\frak b}_k = (F_1,...,F_d,f_{k,1},...,f_{k,n})$ has $F_1,...F_d,f_{k,1},...,f_{k,n}$ exactly as the generators of ${\frak a} + {\frak b}_k$. Namely if we denote by $\mu({\frak a} + {\frak b}_k)$ the number of generators of ${\frak a} + {\frak b}_k$, $\mu({\frak a} + {\frak b}_k)=d + n$.
$A/({\frak a} + {\frak b}_k)$ is a Gorenstein local ring of(of dimension $0$).
