Superstability, noetherian rings and pure-semisimple rings,Annals of Pure and Applied Logic

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Superstability, noetherian rings and pure-semisimple rings
Annals of Pure and Applied Logic ( IF 0.8 ) Pub Date : 2020-11-05 , DOI: 10.1016/j.apal.2020.102917
Marcos Mazari-Armida

We uncover a connection between the model-theoretic notion of superstability and that of noetherian rings and pure-semisimple rings.

We characterize noetherian rings via superstability of the class of left modules with embeddings.

Theorem 0.1

For a ring R the following are equivalent.

(1): R is left noetherian.
(2): The class of left R-modules with embeddings is superstable.
(3): For every $λ \geq | R | + ℵ_{0}$ , there is $χ \geq λ$ such that the class of left R-modules with embeddings has uniqueness of limit models of cardinality χ.
(4): Every limit model in the class of left R-modules with embeddings is Σ-injective.

We characterize left pure-semisimple rings via superstability of the class of left modules with pure embeddings.

Theorem 0.2

For a ring R the following are equivalent.

(1): R is left pure-semisimple.
(2): The class of left R-modules with pure embeddings is superstable.
(3): There exists $λ \geq {(| R | + ℵ_{0})}^{+}$ such that the class of left R-modules with pure embeddings has uniqueness of limit models of cardinality λ.
(4): Every limit model in the class of left R-modules with pure embeddings is Σ-pure-injective.

Both equivalences provide evidence that the notion of superstability could shed light in the understanding of algebraic concepts.

As this paper is aimed at model theorists and algebraists an effort was made to provide the background for both.

中文翻译：

超稳定性，noetherian环和纯半环

我们揭示了超稳定性的模型理论概念与noetherian环和纯半简单环之间的联系。

我们通过带有嵌入的左模块类的超稳定性来表征noetherian环。

定理0.1

对于环R，以下含义相同。

（1）: R保留noetherian。
（2）: 带有嵌入的左R-模块的类是超稳定的。
（3）: 对于每个 $λ \geq | [R | + ℵ_{0}$ ，有 $χ \geq λ$ 因此，带有嵌入的左R-模的类具有基数χ的极限模型的唯一性。
（4）: 带有嵌入的左R-模块类别中的每个极限模型都是Σ-内射的。

我们通过带有纯嵌入的左模块类的超稳定性来表征左纯半环。

定理0.2

对于环R，以下含义相同。

（1）: R保留为纯半简单。
（2）: 具有纯嵌入的左R-模块的类是超稳定的。
（3）: 那里存在 $λ \geq {（ | [R | + ℵ_{0} ）}^{+}$ 因此，带有纯嵌入的左R-模的类具有基数λ的极限模型的唯一性。
（4）: 带有纯嵌入的左R-模块类别中的每个极限模型都是Σ-纯内射。