论文标题
降低性指数和可还原性指数
Reducibility index and sum-reducibility index
论文作者
论文摘要
让$ r $为noetherian戒指。对于有限生成的$ r $ -Module $ m $，诺斯科特引入了$ m $的降低性指数，这是出现在$ m $中的s子模块中$ 0 $ 0 $ 0 $ 0的不可减免的分解中的数量。另一方面，对于Artinian $ r $ -Module $ a $，MacDonald证明，出现在不redentuction $ a $ a $的不可用的汇总表示中的汇总s子模型的数量并不取决于表示的选择。此数字称为$ a $的总可还原索引。在本文的前一部分中，我们计算$ s \ otimes_r m $的可降低性索引，其中$ r \ to s $是Noetherian Rings的平坦同构。特别是，研究了本地化，多项式扩展和$ r $的完成。对于本文的后半部分，我们澄清了$ m $的可降低性指数，$ m $的完成的关系以及$ m $的Matlis Dual的总和还原指数。
Let $R$ be a Noetherian ring. For a finitely generated $R$-module $M$, Northcott introduced the reducibility index of $M$, which is the number of submodules appearing in an irredundant irreducible decomposition of the submodule $0$ in $M$. On the other hand, for an Artinian $R$-module $A$, Macdonald proved that the number of sum-irreducible submodules appearing in an irredundant sum-irreducible representation of $A$ does not depend on the choice of the representation. This number is called the sum-reducibility index of $A$. In the former part of this paper, we compute the reducibility index of $S\otimes_R M$, where $R\to S$ is a flat homomorphism of Noetherian rings. Especially, the localization, the polynomial extension, and the completion of $R$ are studied. For the latter part of this paper, we clarify the relation among the reducibility index of $M$, that of the completion of $M$, and the sum-reducibility index of the Matlis dual of $M$.
