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.
(..) You think I was wrong to cast him out? Five hours later, he's still on your doorstep. There's a strength to him. Stubbornness, arrogance, ambition, I've seen it all before. He reminds you of Kaecilius. I cannot lead another gifted student to power, only to lose him to the darkness. (..)2016 Doctor Strange