Journal of Algebra and Its Applications ( IF 0.8 ) Pub Date : 2021-04-23 , DOI: 10.1142/s0219498822501523 Tran Nguyen An 1 , Tran Duc Dung 2 , Shinya Kumashiro 3 , Le Thanh Nhan 2
Let be a commutative Noetherian ring. For a finitely generated -module , Northcott introduced the reducibility index of , which is the number of submodules appearing in an irredundant irreducible decomposition of the submodule in . On the other hand, for an Artinian -module , Macdonald proved that the number of sum-irreducible submodules appearing in an irredundant sum-irreducible representation of does not depend on the choice of the representation. This number is called the sum-reducibility index of . In the former part of this paper, we compute the reducibility index of , where is a flat homomorphism of Noetherian rings. Especially, the localization, the polynomial extension, and the completion of are studied. For the latter part of this paper, we clarify the relation among the reducibility index of , that of the completion of , and the sum-reducibility index of the Matlis dual of .
中文翻译:
可还原性指数和和-可还原性指数
让是一个可交换的诺特环。对于一个有限生成-模块, Northcott 引入了可还原性指数,这是子模块的冗余不可约分解中出现的子模块的数量在. 另一方面,对于 Artinian-模块, Macdonald 证明了出现在一个冗余和不可约表示中的和不可约子模的数量不依赖于表示的选择。这个数字称为总和可约性指数. 在本文的前一部分,我们计算了可还原性指数, 在哪里是 Noetherian 环的平面同态。特别是定位、多项式扩展和完成被研究。对于本文的后半部分,我们阐明了可还原性指标之间的关系, 的完成, 和 Matlis 对偶的和可约指数.