1. Let phi: R->S be a ring homomorphism, and I an ideal of S. Let J = phi^{-1}(I) := {r in R : phi(r) in I}. Show J is an ideal of R.
2. Let R = C[x_1..x_n]. For each i in 1..n, and t a nonzero complex number, define the ring homomorphism phi_{i,t} : R -> R by phi_{i,t}(x_j) = x_j for j not equal to i, and phi_{i,t}(x_i) = tx_i. Also phi_{i,t}(c) = c for c in C. (From here, you should be able to figure out what phi_{i,t} does to an arbitrary element of R.)
Let I be a monomial ideal. Show that phi_{i,t}^{-1}(I) = I for every i,t.
3. Harder: assume that phi_{i,t}^{-1}(I) = I for every i,t. Show I is a monomial ideal.
Hint: if g is a generator of I, then phi_{i,t}(g) in I for every i,t. Use this to show that g's monomials are also in I.
4. Fix i, and call an ideal I of R[x_1..x_n] x_i-homogeneous if phi_{i,t}^{-1}(I) = I for every t. Show that if I is x_i-homogeneous, then radical(I) is also x_i-homogeneous.
5. Let I be a monomial ideal. Show that radical(I) is also a monomial ideal.
6. Describe an algorithm that, given a system of generators of a monomial ideal, computes the radical (by giving a system of generators).
7. Prove your algorithm works.
