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**Extra resources for Computational Commutative Algebra**

**Example text**

Exercise 8. Let B ⊂ Tn be such that no element in B is divisible by another element in B . Prove that B is finite. Exercise 9. Let (Γ, ◦) be a monoid, and let Σ be a Γ -monomodule. We say that Σ is a Noetherian Γ -monomodule if every ascending chain of Γ -submonomodules Σ1 ⊆ Σ2 ⊆ · · · of Σ is eventually stationary. 4. b) Let Γ be a Noetherian monoid, and let Σ be a finitely generated Γ -monomodule. Then show that Σ is Noetherian. c) Conclude that the Tn -monomodule Tn e1 , . . , er is Noetherian.

1 ................ x1 a) Show that the complement Λ of a monoideal in a monoid is characterized by the following property: if γ ∈ Λ and γ | γ , then γ ∈ Λ . 8. Show that ∆(I) is finitely cogenerated and find a minimal set of cogenerators. c) Now let J = (x51 , x31 x2 , x1 x22 ), and let ∆(J) be the associated monoideal in T2 . Find a set of cogenerators and show that J is not finitely cogenerated.

X1 x22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x31 x2 • 1 . . . . . . . . . . . . . . . . . . . . . .