Question:
I want to determine when the ideals of $\mathbb{Z}/n\mathbb Z$ as $\mathbb{Z}/n\mathbb Z$-modules are projective/injective modules.
Here are some of the cases:
Proj:
(0) $\mathbb{Z/nZ}$ itself is projective $\mathbb{Z/nZ}$-module;
(1) $\mathbb{Z/nZ}$ is local ring iff $n=p^r$. (See Show that $\mathbb{Z}_n$ is local ring iff $n$ is a power of a prime number).
Since projective is equivalent to free over a local ring, and ideal $I=\mathbb{{p^sZ}\over{p^rZ}}$ is not free ($p^{r-s}\in ann(I)$), all nontrivial ideals of $\mathbb{Z/p^rZ}$ are not projective;
(2) $\mathbb{Z/nZ}\simeq \mathbb{Z/p_1Z\times\dots\times Z/p_rZ}$ is a semisimple ring, every module is projective and injective.
Inj:
(0) $\mathbb{Z/nZ}$ itself is injective $\mathbb{Z/nZ}$-module (Baer's criterion);
(1) $\mathbb{Z/p^rZ}$:also apply Baer's criterion. If ideal $I=\mathbb{{p^sZ}\over{p^rZ}}$ is injective, then we can extend $1_I$ to $f:R\to I$. Suppose $f(1+\mathbb{p^rZ})=kp^s+\mathbb{p^rZ})$, then $kp^{2s}=p^s+tp^r$, a contradiction.
Therefore, all nontrivial ideals of $\mathbb{Z/p^rZ}$ are not injective;
(2) $\mathbb{Z/nZ}\simeq \mathbb{Z/p_1Z\times\dots\times Z/p_rZ}$ is a semisimple ring, every module is projective and injective.
But the general cases are unsolved. Can you help me with that? Thanks in advance!
Best Answer
It's known that $\mathbb Z/n\mathbb Z$ is always a quasi-Frobenius ring, and since the injective modules coincide with the projective modules, solving one problem solves the other.
Considering that $R$ is self-injective, we can say that an ideal is injective iff it is a summand of $R$. So I would be comfortable with characterizing the injective/projective ideals of such rings as "the ones generated by idempotents."
The idempotents of $\mathbb Z/n\mathbb Z$ are not difficult to enumerate after expressing $\mathbb Z/n\mathbb Z=\prod \mathbb Z/p_i^{e_i}$ for various primes $p_i$ with exponents $e_i$.