寻求使模满足唯一最大可除子模分解定理的弱于Dedekind环的条件

Awesome question! Let's break this down clearly, starting with a quick recap of Kaplansky's original theorem to set the stage:

Dedekind环上的任意模$M$都具有唯一的最大可除子模$D$；且$M$可分解为$D\oplus E$，其中$E$不含任何可除子模。

Yep, there are definitely ring conditions that are weaker than being a Dedekind ring but still make this theorem hold. Here are the most relevant ones you should know about:

1. Prüfer Domains #

These are a strict generalization of Dedekind rings—every Dedekind ring is a Prüfer domain, but not the other way around (Prüfer domains don't require the Noetherian condition that Dedekind rings mandate). Over Prüfer domains, exactly the same result holds: every module has a unique maximal divisible submodule, which splits off as a direct summand, and the complementary summand has no non-trivial divisible submodules at all.

The key reason this works is that over Prüfer domains, divisible modules are injective. This mirrors a critical property of Dedekind rings, and it's what guarantees the maximal divisible submodule can be cleanly split from the original module. Uniqueness follows from maximality combined with injectivity: any divisible submodule must be contained within the maximal one, so there can't be two distinct maximal divisible submodules.

2. Left Hereditary Rings (Non-Commutative Generalization) #

If you're looking beyond commutative domains, left hereditary rings (rings where every left ideal is projective) satisfy a parallel property for left modules. For these rings, every left module has a unique maximal divisible submodule (defined for non-commutative contexts as modules where multiplying by any regular element $r \in R$ returns the full module: $rM = M$), which acts as a direct summand. The complementary part of the sum has no non-trivial divisible submodules. Since Dedekind rings are commutative hereditary domains, this class is significantly broader.

3. Semi-Hereditary Rings (Commutative, Even Weaker) #

For commutative rings, semi-hereditary rings (where every finitely generated ideal is projective) are another step down in strictness. Prüfer domains are exactly commutative semi-hereditary domains, so in that case, the original theorem's conclusions hold fully. For semi-hereditary rings that aren't domains, divisible modules behave a bit more nuancedly, but for the domain subclass, you're still covered.

Key Takeaway #

At its core, the property that enables this decomposition is that divisible modules are injective over the ring. This holds for Dedekind rings, Prüfer domains, and left hereditary rings—all of which are more general (weaker) than Dedekind rings. Zorn's lemma ensures the existence of a maximal divisible submodule, and injectivity guarantees it splits off as a direct summand. Uniqueness comes from the fact that any divisible submodule must be contained within the maximal one, so there's only one such maximal submodule.

内容的提问来源于stack exchange，提问作者S Ali Mousavi