Air taxi flight crew reported cabin depressurization in cruise flight. Diverted and landed uneventfully.

Narrative

Leveled at 43;000 feet through ZZZ after XA:48 of flight XR:31z we had a depressurization with a rate of 2;000 feet per minute; which is why we did the emergency procedure that marks the QRH that takes us to descend to the MEA; or 10;000 whichever is higher. The first action that was taken was to initiate a 90-degree left turn and descend to FL200. While descending we activated the emergency code since we initially did not contact ATC to resolve the emergency.Approximately through 20;000 feet we had control with the pressurization and slowed the descent; we let ZZZ center know that we were under control. We declared 2 souls on board with 3 hrs of fuel on board; requesting to divert initially to ZZZ1; but after analyzing we realized that ZZZ2 would be a better option. The center asked us if we were still [requesting priority]; which we said no; since we were able to control the cabin pressurization by that moment. At all times we were asked if we required any help or assistance; even before landing; which fortunately was not necessary.When we landed; looking at the plane's system computer; we realized that the out flow valve had failed.It leaves me learning the importance of having an alternate airport in route as well as what it means to be well trained to have a quick and effective response.