Profile updated
I don't need an MX record because I don't use Zimbra as a real mail server, it's just a collaboration software that we use to backup mails and share documents and informations. That said: 1) The DNS is configured and working fine, in fact users can send and receive messages internally 2) No problem to send outside of the domain as well, although I configured Zimbra to use a smarthost and not the DNS.
The problem is that Zimbra thinks that example.com should be resolved internally when a messages to
user@example.com is sent... while some users of example.com are not inside the Zimbra domain