Some factorization properties of \(A+XB[X]\) domains (Q2785915)

Let \(A\subseteq B\) be an extension of integral domains. The integral domain \(R=A+XB[X]= \{f(X)\in B[X]\mid f(0)\in A\}\) is a useful source of examples. This paper deals with the transfer of several factorization properties among the domains \(A,B\) and \(R\). The factorization properties considered are a domain satisfying the ascending chain condition on principal ideals, being atomic (every nonzero nonunit is a product of irreducible elements), being half-factorial or a HFD (atomic plus two factorizations of a nonzero nonunit into irreducibles have the same length), being a finite factorization domain or a FFD (atomic plus a nonzero nonunit has only finitely many nonassociate irreducible factors), and being a GCD-domain. Two results proved, typical of the many results given, are (1) if \(A\) is a field and \(B\) is a UFD, then \(R\) is a HFD and (2) if the quotient field \(\text{qf}(A)\) of \(A\) is contained in \(B\), then \(R\) is a GCD-domain if and only if \(B=\text{qf}(A)\) and \(A\) is a GCD-domain.NEWLINENEWLINEFor the entire collection see [Zbl 0855.00015].