Almost splitting sets in integral domains
Let \(S\) be a saturated multiplicatively closed subset of an integral domain \(A\) and \(N(S)= \{0\neq x\in A\mid (x,s)_v= A\) for all \(s\in S\}\). Then \(S\) is called an almost splitting set if for each \(0\neq d\in A\), there is an integer \(n= n(d)- 1\) such that \(d^n= st\) for some \(s\in S\) and \(t\in N(S)\). In this paper, the author studies almost splitting sets and domains of the form \(A+ X^2B[X]\), where \(B\) is an overring of \(A\). He shows that an almost splitting set \(S\) is a \(t\)-complemented \(t\)-splitting set (but not conversely), and that for an almost splitting set \(S\), the (\(t\)-)class group \(\text{Cl}(D)\) is torsion if and only if \(\text{Cl}(D_S)\) and \(\text{Cl}(D_{N(S)})\) are torsion. He also shows that \(D= A+ X^2B[X]\) is an AGCD-domain if and only if \(R= A+ XB[X]\) is an AGCD-domain and \(\text{char}(A)\neq 0\). (Recall that \(A\) is an AGCD-domain if for every \(a, b\in A\), there is an integer \(n= n(a,b)\geq 1\) such that \(a^nA\cap b^nA\) is principal).
scientific article; zbMATH DE number 2157529
scientific article