Mathematics 894, Problem Set 1

Due January 18, 2008. Read Section 9.6 and Appendix I, section 2. Throughout, F is a field. You may assume that the monomials in F [X_{1}, … x_{n}] form a basis of F [X_{1}, … X_{n}] as a vector space over F.

1. Let ≥ be a total order on the monomials in F [X_{1}, …, X_{n}] satisfying the condition (a) if m_{1} ≥ m_{2} and m is any monomial then mm_{1} ≥ mm_{2}. Prove that conditions (b1) and (b2) are equivalent: (b1) ≥ is a well ordering (b2) m ≥ 1 for all monomials.

2. A monomial order then can be defined either as an order on the monomials in F [X_{1}, …, x_{n}] satisfying (a) and (b1) or as an order satisfying (a) and (b2). Verify that the lexicographic ordering is a monomial ordering.

3. Let I =< m_{1}, …, m_{r} > be an ideal in F [x_{1}, …, x_{n}] generated by monomials m_{i}.

(a) Verify that a monomial m is in I if and only if m is a multiple of one of the m_{i} (i.e. there exists a monomial m′ and i, 1 ≤ i ≤ r such that m = m′m_{i}).

(b) Let I be a monomial ideal in F [X_{1}, … X_{n}]. Let f ∈ F [X_{1}, … X_{n}]. Prove that f ∈ I if and only if each monomial term of f is in I.

4. Prove that in F [X, Y], < X, Y >^{n}=< X^{n}, X^{n−1}Y, …, XY^{n−1}, Y^{n} >, and that all n + 1 generators are required.

5. Show that in the lexicographic order on the monomials of K[X, Y] there are an infinite number of elements between X and Y (even though one cannot have an infinite decreasing sequence of monomials).

Notation: Let R be a commutative ring with 1, and f_{i} ∈ R, 1 ≤ i ≤ n. We will use the notation < f_{1}, …, f_{n} > to denote the ideal of R generated by f_{1}, …, f_{n}. This is so that (f_{1}, …, f_{n}) can be reserved for an element of R^{n}.