jquery-ahm is a replacement for $.ajax. With $.ajax, every ajax request needs a callback, resulting in a lot of javascript. $.ahm reduces javascript code by letting the response data define the actual callbacks.
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Ajax.module will AJAXify any Drupal form. This essentially allows for validation, preview, and submission without a page refresh. Ajax.module supports most forms out of the box. Ajax.module provides both an API for module maintainers as well as an admin interface for easy configuration.