Secondly, there’s an uncanny asymmetry when looking at rings. While these concepts seem dual, Artinian implies Noetherian in the context of rings. Thank you for reading!

To wrap up this thread, I want to mention two things. First, this incorrect characterization can be repaired by replacing “finitely generated” with a somewhat dual notion called “finitely cogenerated“.

The core of this characterization failing is that being Artinian is fundamentally different from being finitely generated (note that every factor module of M being finitely generated is equivalent to M being finitely generated).

To wrap up this thread, there’s an uncanny asymmetry when looking at rings. While these concepts seem dual, Artinian implies Noetherian in the context of rings. Thank you for reading!

The core of this characterization failing is that being Artinian is fundamentally different from being finitely generated (note that every factor module of M being finitely generated is equivalent to M being finitely generated).

From this, it immediately follows that M is an Artinian module whose factor modules are not all finitely generated, as M itself is not finitely generated.

Conversely, if we take the localization of ℤ at a prime number p, denoted by ℤₚ, its elements are of the form a/pⁿ with a ∈ ℤ and n ∈ ℕ₀. Now, looking at the ℤ-module M = ℤₚ/ℤ, one can show that every proper submodule of M is cyclic, generated by 1/pⁿ + ℤ for some n ∈ ℕ₀.

Actually, no—both implications fail in the module setting. For example, ℤ as a module over itself is not Artinian (2ℤ ⊇ 4ℤ ⊇ 8ℤ ⊇ … is an infinite descending chain). However, every factor module of ℤ has the form ℤ/nℤ, all of which are even finite except for the trivial case n=0.

Hence, it raises the question: Is Artinian equivalent to every factor module of M being finitely generated? Prove or disprove!

Now, one can informally think of the minimality condition of Artinian modules as every non-empty set of factor modules of M having a maximal element, where we say M/N ≤ M/N' if and only if N ⊇ N'.

A useful characterization of Noetherian and Artinian modules is that every nonempty set of submodules has a maximal or minimal element, respectively. Of particular importance is that a module is Noetherian if and only if every submodule is finitely generated.

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