She finally achieved/realized/fulfilled her 'life's ambition' [=the thing she most wanted to do in her life] when she started her own business.
He has 'ambitions' for an acting career. = He has 'ambitions' to become an actor. = His 'ambition' is to be/become an actor.
With her talent and fierce 'ambition' [=her very strong desire to succeed], she became a very successful actress.
(..) that his virtues would plead like angels, trumpet-tongued against the deep damnation of his taking-off. I have no spur to prick the sides of my intent but only vaulting ambition which o'erleaps itself and falls on the other. We will proceed no further in this business. Was the hope drunk (..)2015 Macbeth
(..) Yet I do fear thy nature. It is too full o' the milk of human kindness to catch the nearest way. Thou wouldst be great. Art not without ambition but without the illness should attend it. To beguile the time, look like the time. Bear welcome in your eye, your hand, your tongue. (..)2015 Macbeth