Strong atoms in monadically Krull monoids
This manuscript studies strong atoms on monadically Krull monoids, with some applications to the ring of integer-valued polynomials on integral domains. Monoids in this paper are considered to be commutative and cancellative. An atom in a monoid is an element that cannot be expressed as a product of two nonunit elements in the monoid. An atom \(a\) is strong if every divisor of a power of \(a\) is also a power of \(a\). A nonunit \(a\) in a monoid \(M\) is said to be prime if whenever \(a\) divides \(b c\), with \(b\) and \(c\) in \(M\), then either \(a\) divides \(b\) or \(c\), and it is said to be primary if whenever \(a\) divides \(b c\) and does not divide \(b\), then it divides some power of \(c\). For an element \(a\) in a monoid \(H\), the set \([\![a]\!]_H\) denotes the smallest divisor-closed submonoid of \(H\) containing \(a\). A property (P) is said to be monadic for \(H\) if (P) holds for \(H\) if and only if (P) holds for \([\![a]\!]_H\) for every \(a\in H\). A monoid \(H\) has property (S) if for every nonunit \(x\) of \(H\) there exists a finite set of strong atoms \(A\) such that (1) \(x^m = w \prod_{a\in A} a^{m_a}\) for some positive integers \(m\), \(m_a\), \(a\in A\), and some unit \(w\), (2) this expression is unique, and (3) if \(y\) is a strong atom dividing a power of \(x\), then \(y\in A\). The author shows that property (S) is monadic. The extraction degree of \(H\) for two elements \(x,y\in H\) is defined as \(\lambda_H(x,y)= \sup\{ m/n : m\in \mathbb{N}_0, n\in \mathbb{N}, x^m \mid_H y^n \}\). The monoid \(H\) is said to be an extraction monoid if for every nonunit \(x\) in \(H\) and every \(y\in H\), there exists a nonnegative integer \(m\) and a positive integer \(n\) such that \(\lambda_H(x,y)=m/n\). The property of being an extraction monoid is also monadic. A monoid \(H\) is locally finitely generated if for every \(a\in A\), \(([\![a]\!])_H)_{red}\) is finitely generated (where the subindex ''red'' stands for reduced, ''units removed''). The author proves that every locally finitely generated monoid is an extraction monoid. The monoid \(H\) is called monadically Krull if \([\![a]\!]_H\) is Krull for all \(a\in H\). The author proves that every monadically Krull monoid fulfills property (S), it is an extraction monoid, and every primary atom is a strong atom. A monadically Krull monoid is factorial if and only if every strong atom is prime. The last section is devoted to the ring of integer-valued polynomials on an integral domain \(D\), \(\operatorname{Int}(D)=\{ f \in K[x] : f(D)\subseteq D\}\), where \(K\) is the quotient field of \(D\). The author proves as a consequence of the results presented in the previous sections a series of corollaries. For instance, if \(D\) is a Krull domain, then \(\operatorname{Int}(D)\) has property (S), it is an extraction monoid, and every primary atom of \(\operatorname{Int}(D)\) is a strong atom. For \(D\) an integral domain, it is shown that \(\operatorname{Int}(D)\) is factorial if and only if \(D\) is factorial and \(\operatorname{Int}(D)=D[x]\), equivalently, \(D\) is a Krull domain and every strong atom of \(\operatorname{Int}(D)\) is prime.
scientific article; zbMATH DE number 7463821