The sequential Cohen-Macaulayness of idealizations
DOI:
https://doi.org/10.56764/hpu2.jos.2025.4.03.34-41Abstract
Let (\(R\), m) be a Noetherian local ring and \(M\) a finitely generated \(R\)-module. The idealization \(R\ltimes M\), introduced by M. Nagata, has become a useful construction in commutative algebra. Recent work has characterized the approximate Cohen–Macaulayness of such idealizations via the length function associated with a good system of parameters. Motivated by these developments, we investigate via the sequential Cohen–Macaulayness of the idealization \(R\ltimes M\). We provide a characterization in terms of the length function with respect to a good system of parameters of the form \((x_1,0),\ldots,(x_r,0)\), where \(r=\dim R\). Furthermore, we provide equivalent conditions for \(R\ltimes M\) to be sequentially Cohen–Macaulay, expressed in terms of the length functions of both \(R\) and \(M\), and their respective dimension filtrations.
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Copyright (c) 2025 Van-Loc Phan , Minh-Son Doan
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