What is the UMTS-PLMN, and how does it relate to UTRAN and the Core Network?

The UMTS-PLMN (Public Land Mobile Network) is one operator's complete mobile network — it contains both the UTRAN (radio access) and the CN (core) as its two halves.

A PLMN is the network run by a single mobile operator (e.g. one national carrier). It is the umbrella term for everything that operator owns to provide mobile service:

        UMTS-PLMN  (one operator)
       ┌───────────┴───────────┐
     UTRAN                     CN
  (radio access)         (core network)

Key points:

• Each operator runs its own PLMN, identified by a MCC + MNC (Mobile Country Code + Mobile Network Code) — the pair encoded in your IMSI that tells the world which network you belong to.
• Your subscription lives in your home PLMN (HPLMN). When you travel and connect to a different operator's network, that is a visited PLMN (VPLMN) — this is exactly what roaming means.
• So the UMTS-PLMN is not a 5th box next to UE/UTRAN/CN — it is the boundary that encloses the operator's UTRAN and CN.

Tip: PLMN ≈ "one carrier's whole network." Roaming = your phone served by someone else's PLMN.

Go deeper:

• Public land mobile network (Wikipedia) — how the MCC+MNC PLMN code identifies one operator's network and how the phone uses an IMSI/PLMN mismatch to detect roaming.