notifications.php

<?php

/**
 * Hide core updates
 */
add_action( 'after_setup_theme', function () {

  // Still show updates to superadmins
  if ( (! current_user_can( 'update_core' )) || is_super_admin() ) {
    return;
  }

  add_action( 'init', create_function( '$a', "remove_action( 'init', 'wp_version_check' );" ), 2 );
  add_filter( 'pre_option_update_core', '__return_null' );
  add_filter( 'pre_site_transient_update_core', '__return_null' );
}, 1 );


/**
 * Disable plugin update notifications
 */
remove_action( 'load-update-core.php', 'wp_update_plugins' );
add_filter( 'pre_site_transient_update_plugins', '__return_null' );



/**
 * Disable inactive plugins update notifications
 */
add_filter( 'transient_update_plugins', function ( $value = '' ) {
  if ((isset($value->response)) && (count($value->response))) {
    $active_plugins = get_option( 'active_plugins' );
    if ($active_plugins) {
      foreach($value->response as $plugin_idx => $plugin_item) {
        if (!in_array($plugin_idx, $active_plugins)) {
          unset($value->response[$plugin_idx]);
        }
      }
    }
    else {
      foreach($value->response as $plugin_idx => $plugin_item) {
        unset($value->response);
      }
    }
  }
  return $value;
});