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](https://en.wikipedia.org/wiki/GCD_domain), then $D(R) \Doteq (R/R^{\times}, \vert)$ is a bounded distributive [lattice](https://en.wikipedia.org/wiki/Lattice_(order)), 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](https://en.wikipedia.org/wiki/B%C3%A9zout_domain), then $D(R)$ is anti-isomorphic to the lattice $L\_{\text{principal}}(R)$ of its principal ideals.



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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)](https://en.wikipedia.org/wiki/Ascending_chain_condition_on_principal_ideals). Moreover, there exist non-[atomic](https://en.wikipedia.org/wiki/Atomic_domain) 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](https://en.wikipedia.org/wiki/Arithmetical_ring) and this is equivalent to say that the localization $R\_{\mathfrak{m}}$ of $R$ at $\mathfrak{m}$ is a [uniserial ring](https://en.wikipedia.org/wiki/Serial_module) for every maximal ideal $\mathfrak{m}$ of $R$. (An arithmetical domain is the same as [Prüfer domain](https://en.wikipedia.org/wiki/Pr%C3%BCfer_domain)). The Jaffard-Ohm-Kaplansky Theorem states that the positive cone of every [lattice-ordered Abelian group](https://en.wikipedia.org/wiki/Partially_ordered_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](https://en.wikipedia.org/wiki/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



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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!



What is known about ideal and divisibility lattices of GCD domains and their generalizations?
Given a ring $R$, let us denote by $L(R)$ the [lattice](https://en.wikipedia.org/wiki/Lattice_(order)) of two-sided ideals of $R$ for which the infimum and supremum are given by $\inf(I, J) = I \cap J$ and $\sup(I, J) = I + J$.


Such lattices are complete and modular. If $R$ is a principal ideal domain (see [original MSE question/answer](https://math.stackexchange.com/questions/2463311/description-of-the-lattice-that-gcd-and-lcm-form-on-a-pid)) or if $R$ is the ring of integers of a number field [3, page 135], then $L(R)$ is distributive.


The following claims classify the lattices of principal ideal domains (PIDs).



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Claim 1 is, I believe, immediate. (In [2], it is proven that $L(\mathbb{Z}[\sqrt{-5}]) \simeq L(\mathbb{Z})$; this should give an idea as how to proceed.)


So are the following claims.



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> > **Claim 2.** Let $\kappa$ be a cardinal number. Then there is a PID with exactly $\kappa$ prime ideals.
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> > *Proof of Claim 2.* For $\kappa = 1$, take $R = \mathbb{Q}$. For $2 \le \kappa < \omega$, consider $\kappa - 1$ distinct prime numbers $p\_1, \dots, p\_{\kappa - 1}$ in $\mathbb{Z}$ and set $S \Doteq p\_1\mathbb{Z} \cup \cdots \cup p\_{\kappa - 1}\mathbb{Z}$. Then the localization $\mathbb{Z}\_S$ of $\mathbb{Z}$ is a PID with exactly $\kappa$ maximal ideals. For any $\kappa \ge \omega$, set $R = k[X]$ for $k$ an algebraically closed field of cardinality $\kappa$. (For $\kappa = \omega$, setting $R = \mathbb{Z}$ is a simpler and natural choice.)
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> > **Claim 3.** The ideal lattice $L(R)$ of a principal ideal domain $R$ is totally ordered if and only if $R$ is a field or [discrete valuation ring](https://en.wikipedia.org/wiki/Discrete_valuation_ring). It is Boolean if and only if $R$ is a field.
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The question as to whether $L(R)$ is **complemented** was settled by Robert Blair in [1, Theorems 1 and 2] in a broader setting:



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> $(iii)$ The ring $R$ is isomorphic with a direct sum of simple rings. 
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The question as to whether $L(R)$ is **distributive** for more general rings $R$ was also tackled by Robert Blair in [1, Section 5]. Here is one of Blair's results.



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> > **Blair's Lemma. (Lemma 11)** If $R$ is a [primitive ring](https://en.wikipedia.org/wiki/Primitive_ring) whose lattice $L\_r(R)$ of right ideals is distributive, then $R$ is a division ring. 
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[1] R. Blair, ["Ideal lattices and the structure of rings"](http://www.ams.org/journals/tran/1953-075-01/S0002-9947-1953-0055974-3/S0002-9947-1953-0055974-3.pdf), 1952.  

 [2] H. Subramanian, ["Principal ideal in the ideal lattice"](http://www.ams.org/journals/proc/1972-031-02/S0002-9939-1972-0284424-4/S0002-9939-1972-0284424-4.pdf), 1972.  

 [3] G. Birkhoff, "Lattice theory", 1948. 


