论文标题
在几类模块中表征分类性
Characterizing categoricity in several classes of modules
论文作者
论文摘要
我们表明，在红衣主教的尾巴中分类的条件可以在几类模块中以代数为特征。 $定理。$假设$ r $是与统一的联想戒指。 1。本地纯纯$ r $ -modules的类别为$λ$ - 分类中的$ $ $ $ $ $ $λ> | r |+\ aleph_0 $，仅当$ r \ r \ cong m_n（d）$ for $ d $ for $ d $ a division Ring and $ n \ n \ geq 1 $。 2。平面$ r $ -modules的类是$λ$ - 分类中的$ $ $λ> | r | + \ aleph_0 $，仅当$ r \ r \ cong m_n（k）$ for $ k $一个本地环，以便其最大理想是$ t $ nilpotent和$ n \ geq 1 $。 3。假设$ r $是一个交换戒指。绝对纯$ r $ -Modules的类是$λ$-分类$ $ $ $λ> | r | + \ aleph_0 $，仅当$ r $是当地的Artinian戒指。 我们表明，在上述结果中，足以假设$λ$ - 分类为$ $大的红衣主教$λ$。这表明Shelah的分类猜想适用于本地纯的注射模块，扁平模块和绝对纯模块的类别。这些类不是用于任意环的一阶公理。 我们提供的环使得扁平模块的类别在红衣主教的尾部中是分类的，但它不是一阶公理。
We show that the condition of being categorical in a tail of cardinals can be characterized algebraically for several classes of modules. $Theorem.$ Assume $R$ is an associative ring with unity. 1. The class of locally pure-injective $R$-modules is $λ$-categorical in $all$ $λ> |R|+\aleph_0$ if and only if $R \cong M_n(D)$ for $D$ a division ring and $n \geq 1$. 2. The class of flat $R$-modules is $λ$-categorical in $all$ $λ> |R| + \aleph_0$ if and only if $R \cong M_n(k)$ for $k$ a local ring such that its maximal ideal is left $T$-nilpotent and $n \geq 1$. 3. Assume $R$ is a commutative ring. The class of absolutely pure $R$-modules is $λ$-categorical in $all$ $λ> |R| + \aleph_0$ if and only if $R$ is a local artinian ring. We show that in the above results it is enough to assume $λ$-categoricity in $some$ large cardinal $λ$. This shows that Shelah's Categoricity Conjecture holds for the class of locally pure-injective modules, flat modules and absolutely pure modules. These classes are not first-order axiomatizable for arbitrary rings. We provide rings such that the class of flat modules is categorical in a tail of cardinals but it is not first-order axiomatizable.
