By Spiegel M.R.

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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 .

Then it is clear that f is primitive. Suppose we have in Q(R)[x] an equation f = g1 h1 with non-zero and non-invertible polynomials g1 , h1 ∈ Q(R)[x]. Then g1 and h1 are of positive degree. By possibly clearing the denominators, we see that there exists an 34 1. Foundations element r ∈ R such that rf = g2 h2 with g2 , h2 ∈ R[x]. 10 we know that r = cont(g2 ) · cont(h2 ). Thus we can simplify and get a new equation f = g3 h3 with primitive polynomials g3 , h3 ∈ R[x]. Since the degrees of g3 and h3 are positive and R is an integral domain, neither is a unit, contradicting the irreducibility of f .

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.