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 $$0 \le r \le k$$. 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