When a door jam cracks because someone forced the door it is often easier and just as sturdy to repair it rather than to replace it. To replace it requires removing and reinstalling the entire frame.
Find screws long enough to reach at least one inch past the crack from one side or the other. 4 inch screws are often used, and often very thick, so you must drill two holes to avoid cracking the frame with the screws. Drill one hole that is big enough to let the screw slip easily in. This must reach the crack. Then drill a smaller hole as a pilot hole that goes the full length of the screw. You probably cannot get the really proper size in an extra long drill, but you can fairly easily find long drill bits 1/8th of an inch in diameter and this will do.
Use a injector of some kind to fill the crack with glue when it is forces slightly open. Many things like teeth whitening gel come in plastic injectors that make great glue guns.
Now carefully drive in the screw, one about every 6 inches. When the glue dries, this frame will actually be stronger than the original.