Duplocates Records Not Allowing using Triggers and cant change field name

trigger DuplicatesTask on TStudent__c  (before insert,before update,before delete,
                             after insert,after update,after delete) {
       if(trigger.IsInsert) {
              for(TStudent__c  o:[select name from TStudent__c ]){
                   for(TStudent__c  v:trigger.new){
                         if(v.name==o.name){
                               //v.adderror('noduplications');
                               Trigger.new[0].name.addError('Duplicate names does not allow');
                          }
                    }
                }   
           }  //end of else part
       if(trigger.Isupdate){
             for(TStudent__c o:[select Email_Id__c from TStudent__c]){
               for(TStudent__c mv:trigger.new){
                   if(mv.Email_Id__c!=o.Email_Id__c){
                       mv .Email_Id__c.addError('Email Id cant be changed');//Trigger.new[0]
                   }                 
               }
             } 
           }    
}