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

Statement
Suppose $$L$$ is a nilpotent ring and $$I$$ is an ideal of $$L$$. Suppose $$p^r$$ is a prime power dividing the order of $$I$$. Then, the number of ideals of $$L$$ that have order $$p^r$$ and are contained in $$I$$ is congruent to 1 mod $$p$$.

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