Nagata rings and directed unions of Artinian subrings (Q1774373)

Throughout \(R\) is a commutative ring with identity. For \(f\in R[X]\), denote by \(\sigma(f)\) the ideal of \(R\) generated by the coefficients of \(f\). Then \(S=\{f\in R[X]|\, \sigma(f)=R\}\) is a multiplicatively closed subset of \(R[X]\) and the localization \(R(X)=S^{-1}R[X]\) is called the Nagata ring (in one variable) over \(R\). The author studies when such a ring is a directed union of Artinian subrings. Also, it is showed that, for a family of zero-dimensional rings \((R_{\alpha})_{\alpha \in A}\), \(\prod_{\alpha \in A}R_{\alpha}(X_{\alpha})\) is not a directed union of Artinian subrings.