Regularity of $\mathbb Z/ n \mathbb Z$.

Question

For which values $n$ is the ring $\mathbb Z/ n\mathbb Z$ regular? We know that when $n$ is prime, this is regular. When $n$ is not square-free, it's not regular because it is not reduced. However, what if $n$ is square-free?

Answer 1

Suppose $n$ is squarefree and let $p_1 \cdots p_n$ be a prime factorization of $n$. The Chinese Remainder Theorem implies $\mathbb{Z}/n\mathbb{Z} \cong \prod_{i=1}^n \mathbb{Z}/p_i\mathbb{Z}$ as rings. Each $\mathbb{Z}/p_i\mathbb{Z}$ is a field and thus is regular. Products of regular rings are regular, and so $\mathbb{Z}/n\mathbb{Z}$ is regular.