\(m\)-formally Noetherian/Artinian rings
Let \(R\) be a commutative ring and \(m\) is a positive integer. \(R\) is called \(m\)-formally Noetherian (resp., \(m\)-formally Artinian) if for every increasing (resp., decreasing) sequence \((I_n)_{n \geq 0}\) of ideals (resp., proper ideals) of \(R\), the increasing (resp., decreasing) sequence \((\sum_{i_1+\cdots+i_m=n}I_{i_1}\dots I_{i_m})_n\geq 0\) stablizes. In the paper under review, it is proved that many properties of Noetherian (resp., Artinian rings) are also true for \(m\)-formally Noetherian (resp., \(m\)-formally Artinian) rings. Some of interesting results are as follows: \begin{itemize} \item \(R\) is a \(m\)-formally Noetherian ring if and only if the \(m\)-power of every radical ideal of \(R\) is finitely generated. \item If \(R\) is a \(m\)-formally Noetherian ring, then every regular ideal of \(R\) is finitely generated. \item If \(R\) is a \(m\)-formally Noetherian ring, then the complete integral closure and the integral closure of \(R\) coincides. \item Let \(T\) be a ring that is finitely generated as module over a subring \(R\) with \(\mathrm{dim}\ R=0\). Then \(R\) is \(m\)-formally Noetherian for some \(m\) if and only if \(T\) is \(m^ \prime\)-formally Noetherian for some \(m^ \prime\). \item \(R\) is \(m\)-formally Artinian if and only if for every decreasing sequence \((I_n)_{n\geq 0}\) of proper ideals of \(R\), the decreasing sequence \((I_0^{m-1}I_n)_{n\geq 0}\) stabilizes. \item \(R\) is \(m\)-formally Artinian for some \(m\) if and only if \(R\) is \(m^ \prime\)-formally Noetherian for some \(m^ \prime\) with zero Krull dimension. \end{itemize}
scientific article; zbMATH DE number 7029756
scientific article