Congruence condition on number of ideals with quotient in a specific variety in a nilpotent ring

In a nilpotent ring of prime power order
Suppose $$\mathcal{V}$$ is a subvariety of the variety of rings, $$p$$ is a prime number, and $$L$$ is a nilpotent ring of order $$p^k$$. Suppose $$0 \le r \le k$$.

Let $$\mathcal{S}$$ be the collection of ideals $$I$$ of $$L$$ such that the order of $$I$$ is $$p^r$$ and $$L/I \in \mathcal{V}$$. Then, either $$\mathcal{S}$$ is empty or the size of $$\mathcal{S}$$ is congruent to 1 mod $$p$$.

Similar facts

 * Congruence condition on number of normal subgroups with quotient in a specific variety in a group of prime power order
 * Congruence condition on number of ideals of given prime power order in nilpotent ring