Congruence condition on number of ideals of given prime power order and bounded exponent in nilpotent ring

Statement
Suppose $$L$$ is a finite nilpotent ring. Let $$p$$ be a prime number and $$0 \le d \le r$$ be integers. Let $$\mathcal{S}$$ be the collection of ideals of $$L$$ that have order exactly $$p^r$$ and exponent dividing $$p^d$$. Then, either $$\mathcal{S}$$ is empty or the size of $$\mathcal{S}$$ is congruent to 1 mod $$p$$.

Related facts

 * 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
 * Congruence condition on number of ideals of given prime power order in a given ideal in a nilpotent ring

Facts used

 * 1) uses::Nilpotency is subring-closed
 * 2) uses::Congruence condition on number of ideals of given prime power order in a given ideal in a nilpotent ring

Proof
Given: A nilpotent ring $$L$$, a prime number $$p$$, nonnegative integers $$d \le r$$. $$\mathcal{S}$$ is the collection of ideals of $$L$$ of order exactly $$p^r$$ and exponent dividing $$p^d$$.

To prove: Either $$\mathcal{S}$$ is empty or the size of $$\mathcal{S}$$ is congruent to 1 mod $$p$$.

Proof: