The divisibility relation "$a$ divides $b$", or concisely, $a \vert b$ defined over a commutative integral domain $R$ with identity induces a partial order on the multiplicative semigroup $R/R^{\times}$ where $R^{\times}$ is the unit group of $R$. The inclusion relation $I \subset J$ is a partial order on the set of principal ideals of $R$ and the two corresponding partially ordered sets are anti-isomorphic in a natural sense.

If $R$ is a GCD domain, then $D(R) \Doteq (R/R^{\times}, \vert)$ is a bounded distributive lattice, the joint $a \vee b$ being the lowest common multiple of $a$ and $b$ and the meet $a \wedge b$ being the greatest common divisor of $a$ and $b$. We call $D(R)$ the *divisibility lattice of $R$*.

If $R$ is any ring (commutative or not, with identity or not and possibly with zero divisors), the set $\mathcal{I}(R)$ of ideals of $R$ is partially ordered by the inclusion and the operations $I \vee J \Doteq I + J$ and $I \wedge J \Doteq I \cap J$ turns $L(R) \Doteq (\mathcal{I}(R), \subset)$ into a complete and modular lattice. We call $L(R)$ the *ideal lattice of $R$*.

If $R$ is a principal ideal domain, then $D(R)$ and $L(R)$ are naturally anti-isomorphic. More generally, if $R$ is a Bézout domain, then $D(R)$ is anti-isomorphic to the lattice $L_{\text{principal}}(R)$ of its principal ideals.

Question 1.Are there remarkable results that bind the structure of $R$ to lattice properties of $D(R)$ and $L(R)$? In particular, what can be said about $R$ if one of the two lattices is

$(i)$ complete or, $(ii)$ distributive or, $(ii)$ complemented or, $(iii)$ boolean?

Here are the few things I know. The lattice $D(R)$ is complete if and only if $R$ satisfies the ascending chain condition on principal ideals (ACCP). Moreover, there exist non-atomic Bézout rings, e.g., the ring of entire functions (think about $\sin(\frac{z}{2^k})$ for $k \ge 0$). Hence there exists a ring $R$ which is a GCD domain and such that $D(R)$ is not complete. If the lattice $L(R)$ is distributive for a commutative ring $R$, then $R$ is called arithmetical and this is equivalent to say that the localization $R_{\mathfrak{m}}$ of $R$ at $\mathfrak{m}$ is a uniserial ring for every maximal ideal $\mathfrak{m}$ of $R$. (An arithmetical domain is the same as Prüfer domain). The Jaffard-Ohm-Kaplansky Theorem states that the positive cone of every lattice-ordered Abelian group arises as the semigroup of divisibility of a Bézout domain.

The answers of Sridhar Ramesh and Luc Guyot show that for a unique factorization domain $R$, the lattice $D(R)$ is complemented if and only if $R$ is a field, and $D(R)$ is Boolean in this case. In addition, $D(R)$ is totally ordered if and only if $R$ is a discrete valuation ring or a field.

My second question is

Question 2.Have $D(R)$ and $L(R)$ been studied or classified for specific classes of rings such as principal ideal domains (PIDs), unique factorizations domains (UFDs), Dedekind domains and rings of integers in particular?

The answers of Sridhar Ramesh and Luc Guyot show that the situation is extremely tamed for UFDs, since the cardinality of the set of primes of $R$ (up to multiplication by units of $R$) is a complete invariant of isomorphism for $D(R)$. Indeed, $D(R)$ identifies with the lattice of $\mathbb{N}$-valued functions with finite support on the set of primes (modulo units), plus an additional top element corresponding to $0$. Any cardinal can be achieved as the cardinality of the set of primes of some UFD (think about $k[X_i; i \in I]$ for $k$ a field and any set $I$) and actually it is even true for a PID (think of $k[X]$ for a $k$ an arbitrary algebraically closed field).

Any references, specifically comprehensive surveys or textbooks, will be appreciated!