MAS400: Exercises 4

Throughout this sheet F will denote an arbitrary field unless otherwise stated.

1. Let f_{1} = xy − 1 and f_{2} = y^{2} − 1 be elements of ℚ[x, y]. Show that {f_{1}, f_{2}} is not a Gröbner basis for ⟨f_{1}, f_{2}⟩, with respect to ⩽_{lex} and x > y.

2. Let G = {g_{1}, …, g_{t}} ⊆ F [x_{1}, …, x_{n}] with 0 ∉ G. Suppose that, with respect to a fixed monomial order, we have:

∀f ∈ ⟨g_{1}, …, g_{t}⟩, f rem (g_{1}, …, g_{t}) = 0.

Show that G is a Gröbner basis for ⟨G⟩.

3. Let R be a ring (commutative with 1). Show that each ideal of R is finitely generated if and only if there are no infinite ascending chains of ideals:

I_{0} < I_{1} < I_{2} < I_{3} < · · · .

NB: I have proved a very similar result in lectures.