A unital leftR-moduleRMis said to have property (S) if every surjective endomorphism ofRMis an automorphism, the ringRis called left (right)S-ring if every left (right)R-module with property (S) is Noetherian,Ris calledS-ring if it is both a left and a rightS-ring. In this note we show that a duo ring is a leftS-ring if and only if it is left Artinian left principal ideal ring. To do this we shall construct on every non distributive Artinian local ring with radical square zero a non-finitely generated module with property (S). And we give an example of left duo left Artinian left principal ideal ring which is not a leftS-ring, showing the necessity of the ring to be duo in the above result.