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**Sample text**

R} we define the set Bi = {t ∈ Tn | tei ∈ B} ⊆ Tn . By Dickson’s Lemma, the monomial ideals Ii = (Bi ) have finite systems of generators Gi ⊆ Bi . Obviously the P -module M is then generated by G1 e1 ∪ · · · ∪ Gr er ⊆ Tn he1 , . . , er i. This proves a) and the claim M = P r r i=1 Ii ei in b). The fact that this sum is direct follows from M ⊆ ⊕i=1 P ei . a holds for monomial modules. b is true. 10. Every ascending chain of monomial submodules of P r is eventually stationary. Proof. Suppose there exists a strictly ascending chain M1 ⊂ M2 ⊂ · · · of monomial submodules of P r .

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 0 | γ , then γ 0 ∈ 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.

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