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By Kreuzer and Robbiano

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Proof. Let R be factorial, and let r ∈ R be an irreducible element. Suppose r | ab . Hence we have an equation ab = cr with non-units a, b, c ∈ R \ {0}. Then the irreducible factor r must show up either in the factorization of a or in that of b. Therefore we have either r | a or r | b which shows that r is a prime. Conversely, let a1 a2 · · · as = b1 b2 · · · bt be factorizations of the same element. We see that b1 b2 · · · bt ∈ (a1 ), hence the assumption implies that one of the factors has to be in (a1 ).

Let R be a ring, let n ≥ 1 , let P = R[x1 , . . , xn ] be a polynomial ring, and let r ≥ 1. 7. A P -submodule M ⊆ P r is called a monomial module, if it has a system of generators consisting of elements of Tn e1 , . . , er . A monomial submodule of P is also called a monomial ideal of P . Monomial ideals can be readily visualized, especially when there are just two or three indeterminates. 8. For monomial ideals I ⊆ R[x1 , x2 ], we can illustrate the set of terms in I as follows. A term xi1 xj2 ∈ T2 is represented by the point (i, j) ∈ N2 .

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.

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