 Conversely, if is a graded algebra generated over the field by homogeneous elements of degree 1, then the map which sends onto defines an homomorphism of graded rings from formula_26 onto .Its kernel is a homogeneous ideal and this defines an isomorphism of graded algebra between formula_27 and .Thus, the graded algebras generated by elements of degree 1 are exactly, up to an isomorphism, the quotients of polynomial rings by homogeneous ideals.