4. (10 points) Let k be a field and k[x, y, z] = k 131. Define the quotient ring R = k[x, Y, z]/(xz (y + 1)). (a) Show that R is an integral domain and find the field of fractions K = frac(R). (b) Show that R is a normal ring. Hint: Show that R = k(x)[y] N k(2)[y] in K. (c) Show that R is a UFD if and only if y? + 1 is irreducible over k. Hint: Show that y? + 1 is irreducible iff x is prime. Use Nagata's Criterion for one direction; for the other direction show x is irreducible. (d) The coordinate ring of the sphere S is defined by: A = R[x, y, z]/(x + y + z 1) Show that C OR A = C[r,y, z]/(xz (y + 1)) and conclude that COR A is not a UFD. Note: A is a UFD but this is harder to show.