Ajax object not defined – JS file is in plugin folder

You have registered your JS code in wrong hooks . Always CSS and JS should be registered using wp_enqueue_scripts. Also you have only registered your JS code. You have not enqueued your JS file using wp_enqueue_script( ‘ajax-send-notification-script’ );

function ajax_send_notification_init(){

    wp_register_script( 'ajax-send-notification-script', plugins_url() . '/plugin-name/js/the-plugin.js', array('jquery') ); 

    wp_localize_script( 'ajax-send-notification-script', 'ajax_send_notification_object', array( 
        'ajaxurl' => admin_url( 'admin-ajax.php' )
    ));
    // To enqueque the JS file.
    wp_enqueue_script( 'ajax-send-notification-script' );

}

add_action('wp_enqueue_scripts', 'ajax_send_notification_init');