An -module is Noetherian when every submodule is finitely generated, equivalently when every ascending chain of submodules stabilizes. Quotients and finite direct sums of Noetherian modules are Noetherian.
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In abstract algebra, specifically in the context of module theory, a **Noetherian module** is a module that satisfies the ascending chain condition on its submodules. This means that every increasing chain of submodules eventually stabilizes.