UFD characterization via module theory

Question

While studying module theory, I bumped into some theorems characterizing commutative ring properties through modules :

Let $A$ be a commutative ring with unity, then

1. $A$ is a field iff every module over $A$ is free.
2. $A$ is a PID iff every submodule $N$ of a free module $M$ over $A$ is free with $rk(N)\leq rk(M)$.
3. $A$ is Noetherian iff $\bigoplus\limits_{i\in I} E_i$ is an injective $A$-module for any family $\{E_i\}_{i\in I}$ of injective $A$-modules.
4. $A$ is Artinian iff $\prod\limits_{i\in I} P_i$ is a projective $A$-module for any family $\{P_i\}_{i\in I}$ of projective $A$-modules.
5. $A$ is an integral domain, then $A$ is Dedekind domain iff every submodule of a projective $A$-module is projective.

I noticed that all the ring properties all are strongly linked to the ideals of $A$ (guess because they're the only submodules of $A$).

Is there any other characterization as above, for example for unique factorization domains (UFD)?