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.
(..) Shall we say swords? Pistols at dawn? Have you remembered nothing? Recall the lessons of Adam Smith... ...the father of modern economics. "In competition... "...individual ambition serves the common good." Every man for himself, gentlemen. Those who strike out are stuck with her friends. (..)2001 A Beautiful Mind