I have a classic LAMP platform (Debian, Apache2, PHP5 and MySQL) on a dedicated server.
I heard PHPMailer can send email without having installed SMTP. Is PHPMailer
You can use phpmailer to send using the default php mail() function as well.
I recommend not trying to do things manually using the mail() function, use phpmailer instead and configure it to use mail().
I'd like to point out that even though you're not using an SMTP connection to send the mails yourself, the mail() function will use either an SMTP connection or the server's sendmail program to send out the emails anyways, so that will have to be configured for it to work correctly.