绝对不可分解模
Absolutely Indecomposable Modules
摘要：
如果一个模在论域的每个泛型扩张中都是直接不可分解的，则称它是绝对不可分解的。我们想证明绝对不可分解的大阿贝尔群的存在性。这将遵循一个更一般的结果，关于一个大类的交换环R上的R-模，其中自同态环R在传递到论域的泛型扩张时保持不变。事实证明，在这种情况下，“大”有确切的含义，即小于下面定义的第一个欧米伽-鄂尔多斯基数。我们将首先应用Shelah在1982年建立的具有类似性质的大刚性树上的结果，并证明相关的“R_omega-模”（具有可数个不同子模的R-模）的存在性，最后再讨论R-模。通过R_Ω模有很大的优势，证明变得非常透明，基本上使用一些“线性代数”参数也可供研究生。利用计数变元给出了一般不可分解模的一种新构造。
英文摘要：
A module is called absolutely indecomposable if it is directly indecomposable in every generic extension of the universe. We want to show the existence of large abelian groups that are absolutely indecomposable. This will follow from a more general result about R-modules over a large class of commutative rings R with endomorphism ring R which remains the same when passing to a generic extension of the universe. It turns out that `large' in this context has the precise meaning, namely being smaller then the first omega-Erdos cardinal defined below. We will first apply result on large rigid trees with a similar property established by Shelah in 1982, and will prove the existence of related ` R_omega-modules' (R-modules with countably many distinguished submodules) and finally pass to R-modules. The passage through R_omega-modules has the great advantage that the proofs become very transparent essentially using a few `linear algebra' arguments accessible also for graduate students. The result gives a new construction of indecomposable modules in general using a counting argument.
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