Artinian module

From Academic Kids

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 Z-module Q/Z, the additive group of rational numbers modulo the integers. Every submodule of Q/Z is generated by an elements of the form 1/n, for some positive integer n. For any positive integer n greater than one, the chain <1/n>⊂<1/n2>⊂<1/n3>⊂ ... does not terminate, so Q/Z is not Noetherian. Yet every descending chain of submodules terminates: Each such chain has the has the form <1/n1>⊃<1/n2>⊃<1/n3>⊃... for some integers n1, n2, n3, ..., and the inclusion of <1/ni+1> in <1/ni> implies that ni+1 must divide ni, so n1, n2, n3, ... is a decreasing sequence of positive integers. Thus the sequence terminates, making Q/Z is Artinian.

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