Let S = K[x1, x2, . . . , xn] be a polynomial ring over a fifield K graded by degree. Let R be a graded K-subalgebra of S. Assume that there exists an R-module homomorphism ? : S??R that preserves degrees and restricts to the identity map on R.

(1) Show that R is a direct summand of S as R-module, and describe its complement submodule.

(2) Show that R is a fifinitely generated K-algebra.

Problem 3. Let S = K [x1, x2, . .., In] be a polynomial ring over a field K graded by degree. Let R be a graded K-subalgebra of S. Assume that there exists an R-module homomorphism 7 : S->R that preserves degrees and restricts to the identity map on R. (1) Show that R is a direct summand of S as R-module, and describe its com- plement submodule. (2) Show that R is a finitely generated K-algebra