If "R" is a commutative subring of a left Noetherian ring "S", and "S" is finitely generated as a left "R"-module, then "R" is Noetherian. (In the special case when "S" is commutative, this is known as Eakin's theorem.) However this is not true if "R" is not commutative: the ring "R" of the previous paragraph is a subring of the left Noetherian ring "S" = Hom(Q2, Q2), and "S" is finitely generated as a left "R"-module, but "R" is not left Noetherian.