Nil ideal of the ring of integers modulo $n$

Question

I want a characterization of those positive integers $n$ with prime decomposition $n= {p_1}^{\alpha _1}\cdots {p_t}^{\alpha _t}$ such that the principal ideal of the ring $R=\mathbb Z/n\mathbb Z$ generated by $ p_1\cdots p_ t$ is a subset of the principal ideal generated by ${p_1}^{\alpha _1-1}\cdots { p_ t}^{\alpha _ t-1}$. I just know that the former is the nil ideal of the ring $R$. Thanks for any cooperation!

Answer 1

The ideals $I$ of $ℤ/nℤ$ are in inclusion-preserving one-to-one correspondence to the ideals $J$ of $ℤ$ with $J ⊇ nℤ$. So the question can be formulated within $ℤ$:

For which $n ∈ ℤ$ with prime factorization $n = p_1^{α_1}·…·p_t^{α_t}$ is it true that $$(p_1 ·…·p_t) ⊆ (p_1^{α_1-1}·…·p_t^{α_t-1}),$$ that is $p_1^{α_1-1}·…·p_t^{α_t-1} \mid p_1·…·p_t$ in $ℤ$?

By unique prime factorization, these are those $n$ with $α_1 ≤ 2, …, α_t ≤ 2$.