A (not necessarily commutative) Krull monoid---as introduced by Wauters---is defined as a completely integrally closed monoid satisfying the ascending chain condition on divisorial two-sided ideals. We study the structure of these Krull monoids, both with ideal theoretic and with divisor theoretic methods. Among others we characterize normalizing Krull monoids by divisor theories. Based on these results we give a criterion for a Krull monoid to be a bounded factorization monoid, and we provide arithmetical finiteness results in case of normalizing Krull monoids with finite Davenport constant.

/pdf/non-commutative-krull-monoids-a-divisor-theoretic-approach-33nnhxblow.pdf

Non-commutative krull monoids: a divisor theoretic approach and their arithmetic

Non-commutative Krull monoids: A divisor theoretic approach and their arithmetic

We study the structure of the commutative multiplicative monoid $$\mathbb {N}_0[x]^*$$
 of all the non-zero polynomials in $$\mathbb {Z}[x]$$
 with non-negative coefficients. The monoid $$\mathbb {N}_0[x]^*$$
 is not half-factorial and is not a Krull monoid, but has a structure very similar to that of Krull monoids, replacing valuations into $$\mathbb {N}_0$$
 with derivations into $$\mathbb {N}_0$$
 . We study ideals, chain of ideals, prime ideals and prime elements of $$\mathbb {N}_0[x]^*$$
 . Our monoid $$\mathbb {N}_0[x]^*$$
 is a submonoid of the multiplicative monoid of the ring $$\mathbb {Z}[x]$$
 , which is a left module over the Weyl algebra $$A_1(\mathbb {Z})$$
 .

Factorizations of polynomials with integral non-negative coefficients

Constructions are given of Noetherian maximal orders that are finitely presented algebras over a field K, defined by monomial relations. In order to do this, it is shown that the underlying homogeneous information determines the algebraic structure of the algebra. So, it is natural to consider such algebras as semigroup algebras K[S] and to investigate the structure of the monoid S. The relationship between the prime ideals of the algebra and those of the monoid S is one of the main tools. Results analogous to fundamental facts known for the prime spectrum of algebras graded by a finite group are obtained. This is then applied to characterize a large class of prime Noetherian maximal orders that satisfy a polynomial identity, based on a special class of submonoids of polycyclic-by-finite groups. The main results are illustrated with new constructions of concrete classes of finitely presented algebras of this type.

Primes of height one and a class of noetherian finitely presented algebras

Semigroup Algebras and Noetherian Maximal Orders

On some Subsemigroups of noncommutative krull rings

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P. Wauters

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On some Subsemigroups of noncommutative krull rings