论文标题
理想在交换环中的主要分解，重点是krull环
Prime Factorization of ideals in commutative rings, with a focus on Krull rings
论文作者
论文摘要
令$ r $为具有身份的交换戒指。结构定理说，$ r $是pir（resp。，UFR，ZPI-RING，$π$ -ROIND），并且仅当$ r $是PIDS（分别，UFDS，Dedekind域，$π$ -Domains）和特殊主环的有限直接产品。所有这四种类型的积分域都是Krull域，因此是由结构定理的动机，我们研究了一个环中理想的主要分解，这是Krull域和特殊主环的有限直接产物。这样的戒指将被称为一般的克鲁尔环。众所周知，Krull域的特征是Star Operations $ v $或$ t $，如下所示：当且仅当每个非零适当的$ r $的主体理想都可以写成有限$ v $ - 或$ t $ - $ t $ - $ t $ - $ t $ - prime理想的产品时。但是，对于一般的Krull环而言，情况并非如此。在本文中，我们在$ r $上引入了一个新的星星操作$ u $，因此，$ r $是一般的krull戒指，并且只有在每个适当的主要理想$ r $的理想才能写成有限的$ u $ $ u $ - 主要理想的产品时。我们还研究了一般的Krull环的几种环理论特性，包括Kaplansky型定理，Mori-Nagata定理，Nagata环和Noetherian属性。
Let $R$ be a commutative ring with identity. The structure theorem says that $R$ is a PIR (resp., UFR, general ZPI-ring, $π$-ring) if and only if $R$ is a finite direct product of PIDs (resp., UFDs, Dedekind domains, $π$-domains) and special primary rings. All of these four types of integral domains are Krull domains, so motivated by the structure theorem, we study the prime factorization of ideals in a ring that is a finite direct product of Krull domains and special primary rings. Such a ring will be called a general Krull ring. It is known that Krull domains can be characterized by the star operations $v$ or $t$ as follows: An integral domain $R$ is a Krull domain if and only if every nonzero proper principal ideal of $R$ can be written as a finite $v$- or $t$-product of prime ideals. However, this is not true for general Krull rings. In this paper, we introduce a new star operation $u$ on $R$, so that $R$ is a general Krull ring if and only if every proper principal ideal of $R$ can be written as a finite $u$-product of prime ideals. We also study several ring-theoretic properties of general Krull rings including Kaplansky-type theorem, Mori-Nagata theorem, Nagata rings, and Noetherian property.
