To answer your question: No, document.ready will not fire again once a ajax request is completed. (The content in the ajax is loaded into your document, so there isn’t a second document for the ajax content).
To solve your problem just add the event listener to the Element where you load the ajax content into it.
For example:
$( "div.ajaxcontent-container" ).on( "click", "#id-of-the-element-in-the-ajax-content", function() {
console.log($( this ));
});
For #id-of-the-element-in-the-ajax-content you can use any selector you would use in $("selector"). The only difference is, only elements under div.ajaxcontent-container will be selected.
How it works:
As long as div.ajaxcontent-container exists all elements (if they exist now or only in the future) that match the selector #id-of-the-element-in-the-ajax-content will trigger this click-event.