Let R be a commutative ring. Then R is said to be a regular Noetherian ring provided that any regular ideal is finitely generated. The regular Noetherian properties of polynomial rings are obtained. In particular, an example of a regular Noetherian ring R is given to show that R［x］ need not be regular Noetherian. Then the regular Noetherian properties of amalgamation algebras is studied. Finally, regular Noetherian rings are characterized in terms of regular injective modules and regular coflat modules.