Subring-to-ideal replacement theorem for bounded exponent in nilpotent ring

Statement
Suppose $$L$$ is a nilpotent ring and $$S$$ is a subring of $$L$$ of order $$p^r$$ and exponent dividing $$p^d$$, where $$p$$ is a prime number. Then, $$L$$ contains an ideal of order $$p^r$$ and exponent dividing $$p^d$$.

Related facts

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