Is there an intrinsic lattice-theoretic generalization of the set of principal ideals of a ring?

Question

Let $R, S$ be unital commutative rings.

Upon revisiting basics of ring theory, I've been wondering whether there is an intrinsic lattice-theoretic description of what it means to be a principal ideal.

The set $\DeclareMathOperator{Id}{Id}P\subseteq \Id(R)$ of principal ideals enjoys the following properties:

• Every ideal is the join of the principal ideals it contains
• Minimal ideals are principal
• In general, principal ideals need not be minimal (think $(3)>(6)$ in $ℤ$)
• $P$ need not be downward directed: Considering the product $R[x, y]\oplus S$. Then the non-principal ideal $((x, 0), (y, 0))$ is contained in $((1, 0))$. So not every ideal contained in a principal ideal needs to be principal.

Of course, there is an extrinsic definition of what it means to be principal, noting that $\Id$ forms a closure system in $𝒫(R)$:

Extrinsic definition Let $L$ be a complete meet-semilattice (meet-continuously) embedded in a complete lattice $\tilde L$, i.e. $L$ is a closure system in $\tilde L$. Then an element $l\in L$ is called principal if it is the closure $cl(a) := \bigwedge L\cap (a\uparrow)$ of an atom $a\in \tilde L$.

Question is there an intrinsic axiomatic definition for a set of “principal elements” of a lattice with suitable preconditions (complete, modular, …?) such that

1. $P$ is a set of principal elements in $\Id(R)$
2. perhaps even the only set of principal elements in $\Id(R)$?

(Apologies if my lattice theory is a bit lacking and I make some mistakes in terminology or miss something obvious.)

Answer 1

Eric has already answered this question, but I would like to add a comment concerning the topic in the title Is there an intrinsic lattice-theoretic generalization of the set of principal ideals of a ring?

Claim. (i) If $R$ is a ring and $I=(a)$ is a principal $2$-sided ideal of $R$, then $I$ is a compact element of $\textrm{Id}(R)$. Conversely, (ii) If $L$ is a lattice that is isomorphic to the lattice of $2$-sided ideals of some ring and $c\in L$ is compact, then there exists a ring $R$ and an isomorphism $\varphi\colon L\to \textrm{Id}(R)$ such that $\varphi(c)$ is a principal ideal of $R$.

This shows that compactness is the best approximation to a lattice-theoretic characterization of principalness, at least when discussing single ideals.

Eric already stated that (i) holds in his comments, so let me turn to (ii). Assume that $L=\textrm{Id}(S)$ is the ideal lattice of $S$ for some ring $S$ and that $c\in L$ is compact. By compactness, $c$ must be a finitely generated ideal, say $c = (s_1,\ldots,s_n)$. It is known that every ideal of a matrix ring $R=M_n(S)$ is of the form $M_n(J)$ for some ideal $J$ of $S$. From this it follows that the map $\varphi\colon I\mapsto M_n(I)$ is a lattice isomorphism from $L=\textrm{Id}(S)$ to the ideal lattice $\textrm{Id}(M_n(S))$. Moreover, under the isomorphism $\varphi$ the ideal $c=(s_1,\ldots,s_n)$ maps to $M_n(c)$, which is the principal ideal generated by the diagonal matrix $$C= \begin{bmatrix} s_1 & 0 & \cdots & 0\\ 0 & s_2 & & 0\\ \vdots & & \ddots &\\ 0 & 0 & & s_n \end{bmatrix}. $$ To see why the preceding sentence is true, note that $(C)\in \textrm{Id}(M_n(S))$ is the least ideal of $M_n(S)$ containing $C$, and hence must equal $M_n(I)$ for the least ideal $I$ of $S$ containing the entries of $C$. The least such ideal is $I=c$, so $\varphi(c)$ equals the principal ideal $(C)$.

The above shows that if a lattice $L$ is representable as an ideal lattice, then for any compact $c\in L$ it is possible to represent $L$ as an ideal lattice in such a way that $c$ gets represented by a principal ideal. One might then ask whether there must also exist a representation of $L$ where $c$ gets represented as a non-principal ideal. Here the answer is No. If $c\in L$ is compact and also join irreducible, then $c$ must get represented as a principal ideal in every representation. In particular, if a lattice $L$ is a linear order that is isomorphic to the ideal lattice of a ring, then in every representation of $L$ as an ideal lattice the compact elements of $L$ must get represented by principal ideals.

Answer 2

There is no such description that uniquely characterizes the principal ideals: you can have an isomorphism between lattices of ideals of rings that does not preserve the principal ideals. For instance, consider the case where $R$ is a Dedekind domain. Then (ignoring the zero ideal) $\operatorname{Id}(R)$ can be identified with the lattice of finite support functions $X\to\mathbb{N}$ with the reverse pointwise order where $X$ is the set of nonzero prime ideals, by sending an ideal to its prime factorization. So, any bijection between the nonzero primes in two Dedekind domains $R$ and $S$ induces an isomorphism $\operatorname{Id}(R)\to \operatorname{Id}(S)$. In particular, such a bijection could map a principal prime to a nonprincipal prime, and so the induced isomorphism would not preserve the principal ideals.