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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 reducedsubmodule 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 associatedwith $M$. Interpretations of different reduced modules is given in terms ofMacaulay inverse systems. It is further shown that these reduced submodules areexamples of modules in a torsion-torsionfree class, together with their duals;coreduced modules, exhibit symmetries in regard to Matlis duality and torsiontheories. Lastly, we show that any $R$-module $M$ of the kind described heresatisfies the radical formula.

Reduced submodules of finite dimensional polynomial modules