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

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.)

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.

• This is really nice! I'm curious about what happens if we try to push down to first-order logic: is there a snappy description of the first-order theory of either of the (non-elementary) classes "lattices with a distinguished compact element" or "lattices of ideals with a distinguished compact element" (in each case modulo the theory of lattices with a distinguished element, obviously)? Mar 31, 2022 at 17:19
• It would also be interesting to ask whether the same claim holds if you restrict to commutative rings. Apr 2, 2022 at 21:06

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.

• Interesting, I definitely didn't expect that! Do you know if there's a snappy characterization of the first-order lattice-theoretic properties of prime ideals (that is, the set of first-order sentences in the language of lattices + a constant $c$ that are true in every lattice of ideals of a ring when $c$ is interpreted as a principal ideal)? Mar 30, 2022 at 23:50
• Sorry, "prime" should have been "principal" (I got it right in the parenthetical!). Mar 31, 2022 at 0:13
• I don't know. I would guess that there aren't any interesting properties: that is, the set of properties is the same as the set of properties that are true for an arbitrary element in the lattice of ideals in a ring. Mar 31, 2022 at 0:15
• A nontrivial non-first-order property is that a principal ideal is finitely generated, which is equivalent to saying that it is "compact" (whenever it is a join of ideals, it is a join of a finite subcollection of them). Mar 31, 2022 at 0:16