Homework #2

Here is the next homework assignment:

By the way, below I use (f_1,...,f_s) to denote the ideal generated by the {f_i} instead of the angle brackets, since HTML doesn't like it when I try to use angle brackets.

1.5: 12, 13 (use linearity of the formal derivative, and prove this for monomials first!), 14, 15a
2.2: 2
2.3: 5

1. Let I and J be ideals of k[x_1,...,x_n]. Show that I intersect J, and I+J are ideals, where I+J = {f + g | f is in I and g is in J}.

2. Let I be an ideal of k[x_1,...x_n]. We say I is a monomial ideal if I can be generated by monomials m_1,...,m_s in the variables x_1,...,x_n. For example, (x^2) is a monomial ideal of k[x], while (x+x^2) is not.

a) Show that intersections and sums of monomial ideals are monomial.
b) Let I = (m_1,...,m_s). Compute a generating set for the radical of I, and show that it is also a monomial ideal.
c) Show that a monomial ideal I is radical if and only if I has a monomial generating set where each monomial is squarefree, i.e., the power on each variable in the monomial is 1. (i.e., xyz^2 is not squarefree, but xyz is).

The due date is February 23rd.

Frank