Add and remove attribute with jquery

This is my buttons:

<button id="add">Add</button>
<button id="remove">Remove</button>

this is my JavaScript code:

<script type="text/javascript">
	$(document).ready(function(){	
		/*** Add attribute ***/
		$("#add").click(function(){
			$("#page_navigation1").attr("id","page_navigation1");
		});		
		/*** Remove attribute ***/
		$("#remove").click(function(){
			$("#page_navigation1").removeAttr("id");
		});		
	});
</script>

and this is my html code:

<div id="page_navigation1">next</div>

My problem is, when I click on Remove button everything is fine and ID will be removed, but after click on remove button, when I click on Add button, code is not working and nothing happens!

I need to add and remove ID whith this code, but I can just Remove ID, I cant add ID. I need some help for this, and one more thing,

How I can add CSS style for button when button is Active? like this, when I click on Remove button, I want change button background to red. I need to add css class for add and remove when buttons are active! and I dont know how!

It's because you've removed the id which is how you're finding the element. This line of code is trying to add id="page_navigation1" to an element with the id named page_navigation1, but it doesn't exist (because you deleted the attribute):

$("#page_navigation1").attr("id","page_navigation1");

Demo:

If you want to add and remove a class that makes your <div> red use:

$( '#page_navigation1' ).addClass( 'red-class' );

And:

$( '#page_navigation1' ).removeClass( 'red-class' );

Where red-class is:

.red-class {
    background-color: red;
}

Once you remove the ID "page_navigation" that element no longer has an ID and so cannot be found when you attempt to access it a second time.

The solution is to cache a reference to the element:

$(document).ready(function(){
    // This reference remains available to the following functions
    // even when the ID is removed.
    var page_navigation = $("#page_navigation1");

    $("#add").click(function(){
        page_navigation.attr("id","page_navigation1");
    });     

    $("#remove").click(function(){
        page_navigation.removeAttr("id");
    });     
});

First you to add a class then remove id

<script type="text/javascript">
$(document).ready(function(){   
 $("#page_navigation1").addClass("page_navigation");        
   
 $("#add").click(function(){
        $(".page_navigation").attr("id","page_navigation1");
    });     
    $("#remove").click(function(){
        $(".page_navigation").removeAttr("id");
    });     
  });
</script>

If you want to do this, you need to save it in a variable first. So you don't need to use id to query this element every time.

var el = $("#page_navigation1");

$("#add").click(function(){
  el.attr("id","page_navigation1");
});     

$("#remove").click(function(){
  el.removeAttr("id");
});