How to register script with null value for version?

You literally specify the version number as null:

function add_ajax_script() {
    wp_register_script( 'ajax-implementation', get_stylesheet_directory() . '/scripts/ajax-implementation.js', array('jquery'), null, false);
    wp_enqueue_script('ajax-implementation');
}

add_action( 'wp_enqueue_scripts', 'add_ajax_script' );

An empty string, '' is not the same thing as null.