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
- $P$ is a set of principal elements in $\Id(R)$
- 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.)