Weakly factorial property of a generalized Rees ring \(D[X,d/X]\)
This article investigates the weakly factorial property of a generalized Rees ring \(R:=D[X,d/X]\) where \(D\) is an integral domain, \(d\in D\), and \(X\) is an indeterminate over \(D\). An integral domain \(D\) is said to be a weakly factorial domain (WFD) if each nonzero, nonunit of \(D\) can be written as a finite product of primary elements of \(D\), where \(a\in D\) is primary if \(aD\) is a primary ideal. This article looks to expand on some similar results in the literature about the ring \(S=D[tI, t^{-1}]\) where \(t\) is an indeterminate and \(I\) is an ideal such as: (1) if \(I\) is finitely generated then \(S\) is a UFD if and only if \(D\) is a UFD and \(t^{-1}\) is a prime element of \(S\) [\textit{D. Whitman}, Math. Jap. 17, 13--14 (1972; Zbl 0255.13015)], (2) \(S\) is a GCD-domain if and only if \(D\) is a GCD-domain and \(t^{-1}\) is a prime element of \(S\) [\textit{J. Mott}, Math. Jap. 20, 85--87 (1975; Zbl 0321.13001)] , (3) \(S\) is a Prüfer \(\nu\)-multiplication domain (P\(\nu\)MD) if and only if \(S\) is a (P\(\nu\)MD), \(t^{-1}\) is prime in \(S\) and \(\cap_{n=1}^{\infty} I^n = (0)\) [\textit{H. Kim} et al., Bull. Korean Math. Soc. 41, No. 3, 473--482 (2004; Zbl 1074.13002)]. We note that the situation focused on in this article is achieved when \(I=dD\) and \(X=t^{-1}\). The author is able to prove several nice results including, but not limited to the following: (1) \(R\) is a WFD if and only if \(R\) is a weakly factorial GCD-domain if and only if \(D\) is a weakly factorial GCD domain and \(d=0\), \(d \in U(D)\), or \(d\) is prime in \(D\), (2) \(R\) is a ring of Krull type if and only if \(D\) is of Krull type, and (3) If \(D\) is a weakly factorial GCD-domain, \(p\) is prime in \(D\), and \(n \geq 2\) is an integer, then \(D[X, p^n/X]\) is an almost weakly factorial domain with Cl\((D[X, p^n/X])=\mathbb{Z}_n\). A domain is said to be of Krull type if there is a set \(\{V_\alpha\}\) of valuation overrings of \(D\) such that (1) each \(V_\alpha=D_P\) for some prime ideal \(P\) of \(D\), (2) \(D=\cap V_\alpha\), and (3) this intersection is of finite character. A domain is said to be almost weakly factorial if for each nonzero \(d\in D\), there is an integer \(n=n(d) \geq 1\) such that \(d^n\) can be written as a finite product of primary elements of \(D\). The article is very concisely written and there are diagrams showing the relationship between many of the properties investigated in the article.
scientific article; zbMATH DE number 6999259
scientific article