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arXiv · 2209.13447

Reduced submodules of finite dimensional polynomial modules

Tilahun Abebaw·Nega Arega·Teklemichael Worku Bihonegn·David Ssevviiri
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Abstract

Let $k$ be a field with characteristic zero, $R$ be the ring $k[x_1, \cdots, x_n]$ and $I$ be a monomial ideal of $R$. We study the Artinian local algebra $R/I$ when considered as an $R$-module $M$. We show that the largest reduced submodule of $M$ coincides with both the socle of $M$ and the $k$-submodule of $M$ generated by all outside corner elements of the Young diagram associated with $M$. Interpretations of different reduced modules is given in terms of Macaulay inverse systems. It is further shown that these reduced submodules are examples of modules in a torsion-torsionfree class, together with their duals; coreduced modules, exhibit symmetries in regard to Matlis duality and torsion theories. Lastly, we show that any $R$-module $M$ of the kind described here satisfies the radical formula.

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Tilahun Abebaw, Nega Arega, Teklemichael Worku Bihonegn, David Ssevviiri. 2022-09-27. Reduced submodules of finite dimensional polynomial modules. https://arxiv.org/abs/2209.13447

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