Artinian module

In abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its poset of submodules. They are for modules what Artinian rings are for rings, and a ring is Artinian if and only if it is an Artinian module over itself (with left or right multiplication). Both concepts are named for Emil Artin.

In the presence of the axiom of choice, the descending chain condition becomes equivalent to the minimum condition, and so that may be used in the definition instead.

Like Noetherian modules, Artinian modules enjoy the following heredity property:

The converse also holds:

As a consequence, any finitely-generated module over an Artinian ring is Artinian. Since an Artinian ring is also a Noetherian ring, and finitely-generated modules over a Noetherian ring are Noetherian, it is true that for an Artinian ring R, any finitely-generated R-module is both Noetherian and Artinian, and is said to be of finite length; however, if R is not Artinian, or if M is not finitely generated, there are counterexamples.

The ring R can be considered as a right module, where the action is the natural one given by the ring multiplication on the right. R is called right Artinian when this right module R is an Artinian module. The definition of "left Artinian ring" is done analogously. For noncommutative rings this distinction is necessary, because it is possible for a ring to be Artinian on one side only.

The left-right adjectives are not normally necessary for modules, because the module M is usually given as a left or right R module at the outset. However, it is possible that M may have both a left and right R module structure, and then calling M Artinian is ambiguous, and it becomes necessary to clarify which module structure is Artinian. To separate the properties of the two structures, one can abuse terminology and refer to M as left Artinian or right Artinian when, strictly speaking, it is correct to say that M, with its left R-module structure, is Artinian.

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