A quick cheat sheet:

RequestyesyesEnds after response.
ViewyesnoEnds when user navigates “away”.
SessionyesyesAvailable until session timeout or invalidation.
ApplicationyesyesAvailable until web application is stopped.
ConversationnoyesControlled by conversation.begin() and conversation.end().

JSF @ManagedBean - in javax.faces.bean

EE6 @ManagedBean - in javax.annotation  (not part of JSF)

CDI beans use @Named in javax.inject for injection.  JSF supports rudimentary injection using @ManagedProperty - may be sufficient for your needs, though!

CDI scopes are in javax.enterprise.context

Ajax works well with view-scoped beans (JSF only), but not with request-scoped beans - because the scope does not persist beyond the response phase.