How to override the wordpress default widget markup

I am new to wordpress, Here i use the wordpress TwentyEleven theme, i need to customize the widget markup. It means the suppose i want to change the element ‘ul’ to div and add some class or id to this element. Now currently it was not possible for me because the wordpress default widgets are already available inside wordpress core file..

http://codex.wordpress.org/Function_Reference/unregister_widget

add something like this to functions.php:

if (!function_exists('my_unregister_default_wp_widgets')) {
    function my_unregister_default_wp_widgets() { 
        unregister_widget('WP_Widget_Calendar');
        unregister_widget('WP_Widget_Recent_Posts');
    }
    add_action('widgets_init', 'my_unregister_default_wp_widgets', 1);
}

Also… you can look at the http://codex.wordpress.org/Widgets_API examples to see how you can extend the WP_Widget class to create custom widgets that you want.