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 $p$.

Related facts

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