An example of an atomic pullback without the ACCP
An integral domain is \textit{atomic} if every nonzero nonunit is a product of irreducible elements. \textit{P. M. Cohn} [Proc. Camb. Philos. Soc. 64, 251--264 (1968; Zbl 0157.08401)] observed that a domain satisfying the ascending chain condition on principal ideals (ACCP) is atomic, but incorrectly asserted that the converse was also true. Shortly afterwards, Grams constructed the first example of an atomic domain without the ACCP [\textit{A. Grams}, Proc. Camb. Philos. Soc. 75, 321--329 (1974; Zbl 0287.13002)]. The main purpose of the paper under review is to describe a new method of creating atomic domains without the ACCP, using a novel approach with a pullback construction that is in some ways simpler than previous methods. Even now that the existence of atomic domains without the ACCP is well known, new methods of constructing such domains still hold interest as potential sources of counterexamples to assertions about the atomic property. For instance, \textit{M. Roitman} [J. Pure Appl. Algebra 87, No. 2, 187--199 (1993; Zbl 0780.13014)] produced an example of an atomic domain \(D\) with \(D[X]\) not atomic -- such a domain necessarily fails the ACCP. More recently, the reviewer [Rocky Mt. J. Math. 44, No. 1, 139--155 (2014; Zbl 1296.13003)] adapted Roitman's construction to create an example of an atomic domain that is not a comaximal factorization domain (as defined in [\textit{S. McAdam} and \textit{R. G. Swan}, J. Algebra 276, No. 1, 180--192 (2004; Zbl 1081.13008)]) -- again any such example must fail the ACCP. The following is the specific pullback construction investigated in the paper under review. Let \(D\) be a domain with quotient field \(K \neq D\) and \((v_1),\ldots,(v_r)\) be distinct maximal ideals of \(K[X]\). For \(1 \leq i \leq r\), let \(\theta_i\) be a root of \(v_i\) and \(D_i\) be a non-field overring of \(D[\theta_i]\). Then \(R := \{g(x) \in K[X] \mid \text{each } g(\theta_i) \in D_i\}\) is a domain properly between \(D[X]\) and \(K[X]\) that fails the ACCP. The authors prove necessary and sufficient conditions for \(R\) to be atomic, then use these results to produce a (relatively) simple example of an atomic domain without the ACCP.
scientific article; zbMATH DE number 6950418