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We prove that an associated graded algebra  of a finite dimensional algebra  is  (= selfinjective) if and only if  is  and Loewy coincident. Here  is said to be Loewy coincident if, for every primitive idempotent , the upper Loewy series and the lower Loewy series of  and  coincide. -3 algebras are an important generalization of  algebras; note that Auslander algebras form a special class of these algebras. We prove that for a Loewy coincident algebra , the associated graded algebra...
    			                    
    			                 
    		                
    		                
    		            
    			    
    		            
    		                
    		                
    		                
    			                
    		                
    		                
    		            
    			    			
    			 
 
    			
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