Division with remainder in algebras with valuation (Q646614)

Let \(R\) be a ring with \(1,\) \(A\) an integral \(R\)-algebra, \(K\) the field of fractions of \(A.\) Let \(v:A\setminus \{0\}\rightarrow S\) be a surjective valuation (\(S\) is a totally ordered non-negative abelian semigroup) and \( v:K\rightarrow G(S)\) its unique extension, where \(G(S)\) is the Grothendieck group of \(S\). Let \(W\subset A\setminus \{0\}\) be the subset of \(A\) for which unique division with remainder holds i.e. \(w\in W\) iff 1. the canonical short exact sequence of \(R\)-modules \[ 0\rightarrow (w)\rightarrow A\rightarrow A/(w)\rightarrow 0 \] splits, 2. for each \(a\in A\) \[ a=wq+r \] in \(A\), where \(v(r)<v(w)\) if \(r\neq 0.\) \(A\) is a unique division with remainder domain iff \(W=A\setminus \{0\}.\) The author proves some properties of \(W\) and an analogue of known theorems on characterization of unique factorization domain. His theorem is: An integral domain \(A\) with valuation is unique division with remainder domain iff \(\@(A)=0,\) where \(\@(A):=K^{\ast }/G(W),\) where \(K^{\ast } \) is the topological multiplicative group of topological field \(K\) equipped with the topology of valuation. Some examples of computation of \(\@(A)\) are also given.