On an electronic typewriter, a display cable wraps around a ring. What is it for and is there a way to remove (break) that ring in order to extend the cable length?
It's a ferrite bead. It's a brittle material, so you can definitely break it. Be sure to wear eye protection.
If you want to add a new one later, you can buy beads made for flat cables, like shown below.
That is a ferrite core; its purpose is to suppress EMI. It's there for a reason.
If you want a longer cable (note: you may encounter functional problems if you extend it excessively, as signal quality decreases with cable length), assuming one or both ends are plugged into connectors rather than soldered directly to the boards), it is possible to get IDC connectors. Looks like you need 14 pin. Mouser or DigiKey are good vendors.
You will need to know the precise spacing pin-to-pin for the connectors.
It's a "ferrite bead", which acts as a form of "common mode choke".
Looked at from the perspective of a "common mode" signal where the current only flows through the cable in one direction, the wires wrapped round the ferrite form and inductor, supressing high frequencies.
However looked at from the perspective of a "differential mode" signal, where the signal and it's return wire are both within the same cable, the ferrite ring has little effect.
This is useful, because as a rule wanted signals are differential mode, while interference is common mode. The ferrite bead is only effective over a limited range of frequencies, but it is still useful in reducing both the emission of interence and the susceptibility to interference of the cables within a system.