Some notes on JTTC rings

Abstract A ring R is called JTTC if for any a ∈ N ( R ) and b ∈ R , ( a b ) 2 = a b 2 a , which is a proper generalization of CN rings. In this paper, we show that (1) a ring R is commutative if and only if ( x y ) 2 = x y 2 x for each x ∈ S N ( R ) and y ∈ S Z r ( R ) ; (2) R is a JTTC ring if and only if x y x = x 2 y for each x ∈ N ( R ) and y ∈ S Z r ( R ) ; (3) R is a reduced ring if and only if T 3 ( R ) is a JTTC ring; (4) R is a CN ring if and only if V 2 ( R ) is a JTTC ring; (5) R is a commutative reduced ring if and only if T V 4 ( R ) is a JTTC ring; (6) R is a commutative ring if and only if G 3 ( R ) is a JTTC ring; (7) If R is a JTTC ring containing a von Neumann regular maximal left ideal, then R is a strongly regular ring.