Congruence condition on number of ideals of given prime power order in nilpotent cring of prime power order

Statement

Suppose $C$ is a nilpotent cring of prime power order, i.e., $C$ is a nilpotent cring and its order is a prime power $p^{k}$. Suppose $0\leq r\leq k$. Then, the number of ideals of $C$ of order $p^{r}$ is congruent to 1 mod $p$.

Related facts

• Congruence condition on number of subcrings of given prime power order in nilpotent cring of prime power order
• Congruence condition on number of subrings of given prime power order in nilpotent ring
• Congruence condition on number of ideals of given prime power order in nilpotent ring