Atomicity and the fixed divisor in certain pullback constructions

Jason Greene Boynton¹

¹ Department of Mathematics North Dakota State University Fargo, ND 58108-6050, U.S.A.

Colloquium Mathematicum, Tome 129 (2012) no. 1, pp. 87-97

Résumé

Let $D$ be an integral domain with field of fractions $K.$ In this article, we use a certain pullback construction in the spirit of $\mathop {\rm Int}(E,D)$ that furnishes many examples of domains between $D[x]$ and $K[x]$ in which there are elements that do not admit a finite factorization into irreducible elements. We also define the notion of a fixed divisor for this pullback construction to characterize all of its irreducible elements and those nonzero nonunits that do admit a finite factorization into irreducibles. En route to these characterizations, we show that this construction yields a domain with infinite restricted elasticity.

Détail

DOI : 10.4064/cm129-1-6

Keywords: integral domain field fractions article certain pullback construction spirit mathop int furnishes many examples domains between which there elements admit finite factorization irreducible elements define notion fixed divisor pullback construction characterize its irreducible elements those nonzero nonunits admit finite factorization irreducibles route these characterizations construction yields domain infinite restricted elasticity