Artinian module

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In abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its submodules.

Artinian modules are an analogue of Artinian rings. Both are named after Emil Artin.

When working with an Artinian ring we must distinguish between being left, right, or two-sided Artinian, but this distinction does not make sense when working with modules.

Unlike the case of rings, there are Artinian modules which are not Noetherian modules. For example, consider the p-primary component of $\mathbb{Q}/\mathbb{Z}$ , that is $\mathbb{Z}[1/p] / \mathbb{Z}$ , which is isomorphic to the p-quasicyclic group $\mathbb{Z}(p^{\infty})$ , regarded as $\mathbb{Z}$ -module. The chain $\langle 1/p \rangle \subset \langle 1/p^2 \rangle \subset \langle 1/p^3 \rangle \cdots$ does not terminate, so $\mathbb{Z}(p^{\infty})$ (and therefore $\mathbb{Q}/\mathbb{Z}$ ) is not Noetherian. Yet every descending chain of (without loss of generality proper) submodules terminates: Each such chain has the form $\langle 1/n_1 \rangle \supseteq \langle 1/n_2 \rangle \supseteq \langle 1/n_3 \rangle \cdots$ for some integers $n 1, n 2, n 3,$ ..., and the inclusion of $\langle 1/n_{i+1} \rangle \subseteq \langle 1/n_i \rangle$ implies that $n i + 1$ must divide $n i$ . So $n 1, n 2, n 3,$ ... is a decreasing sequence of positive integers. Thus the sequence terminates, making $\mathbb{Z}(p^{\infty})$ Artinian.