Ajax sending data from other php file contact form

I am new to AJAX, i came from a code in internet about sending Ajax to php file. The problem is i dont know if the data is submitted or what happen. The alert box doesn't pop up if it was submited

success: function () {
              alert('form was submitted');
            }

Here is the full form from html :

    <form class="form-horizontal">
    <div class="form-group">
    <label for="contact-name" class="col-sm-2 control-label">Name</label>
     <div class="col-sm-3">
    <input type="text" class="form-control" id="name" name="name" placeholder="FULL Name" required>
     </div>
     </div>
     <div class="form-group">
  <label for="inputEmail3" class="col-sm-2 control-label">Email</label>
   <div class="col-sm-4">
  <input type="email" class="form-control" id="email" name="email" placeholder="Email" required>
  </div>
  </div>
  <div class="form-group">
  <label for="contact-msg" class="col-sm-2 control-label">Message</label>
  <div class="col-sm-4">
  <textarea name="message" class="form-control" required></textarea>
    </div>
   </div>
   <div class="form-group">
   <div class="col-sm-offset-2 col-sm-10">
  <button class="btn btn-primary" type="submit" value="submit" >Send Message</button>
   </div>
   </div>
  </form>

here is the script :

<script>
      $(function () {

        $('form').on('submit', function (e) {

          e.preventDefault();

          $.ajax({
            type: 'post',
            url: 'mail.php',
            data: $('form').serialize(),
            success: function () {
              alert('form was submitted');
            }
          });

        });

      });
</script>

and because i dont know where the problem is, please check also my php code. thank you stackoverflow,good programmers. php code :

<?php

    $name = $_POST['name']; 
    $email_from = $_POST['email']; 
    $Idea = $_POST['message']; 

    $email_to = "[email protected]";
    $email_subject = "Message from WindSor Website";
    $email_message = "Form details below.";

        function clean_string($string) {
          $bad = array("content-type","bcc:","to:","cc:","href");
          return str_replace($bad,"",$string);
        }

        $email_message .= "Name: ".clean_string($name)."n";
        $email_message .= "Email: ".clean_string($email_from)."n";
        $email_message .= "Comments: ".clean_string($Idea)."n";

        // create email headers
      $headers = 'From: '.$email_from."rn".
      'Reply-To: '.$email_from."rn" .
      'X-Mailer: PHP/' . phpversion();
       mail($email_to, $email_subject, $email_message, $headers);


        echo "<script type='text/javascript'>alert('$message');</script>";        
?>

am i doing right ? or maybe i was doing something that nothing goes.