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

Statement
Suppose $$L$$ is a finite fact about::nilpotent Lie ring and $$p^r$$ is a prime power dividing the order of $$L$$. Then, the number of ideals of $$L$$ of order $$p^r$$ is congruent to $$1$$ modulo $$p$$.

Similar facts

 * Congruence condition on number of subrings of given prime power order in nilpotent Lie ring
 * Congruence condition on number of subgroups of given prime power order

Opposite facts

 * Congruence condition fails for number of subrings of given prime power order

Facts used

 * 1) uses::Congruence condition on number of ideals of given prime power order in nilpotent ring

Proof
The proof follows directly from Fact (1), which is the more general version for arbitrary nilpotent rings (not necessarily commutative, associative, or Lie).