An intersection of principal left ideals need not be principal but incidentally this phenomenon is witnessed in von Neumann regular rings. How about arbitrary intersections of infinitely many principal left ideals in vN-regular rings? Must they be principal? Are there any natural classes of (unital) rings which have such a property?
I know that the central idempotents of a (right or left) self injective VNR ring is complete, so at the very least any commutative, self-injective VNR ring works for you.
I still feel like it is probably false in general. I don't know enough about von Neumann's "continuous geometry", but I feel like the completeness of those lattices have something to do with the completeness of the lattice of principal left ideals of a VNR ring.
Edit OK, I think this paper by Birkenmeier, Kim and Park is what you're looking for. In case you can't access that, the title is Quasi-Baer ring extensions and biregular rings.
The intro mentions that there are regular rings whose lattices of one sided ideals are not complete.