2022/05/24 by Xiaolei Zhang, Zhang, Xiaolei
Mathematics · Neuroscience · #Commutative Algebra and Its Applications #Mind wandering and attention #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2205.11724
In this paper, we say a ring R is Nil∗-Noetherian provided that any nil ideal is finitely generated. First, we show that the Hilbert basis theorem holds for Nil∗-Noetherian rings, that is, R is Nil∗-Noetherian if and only if R[x] is Nil∗-Noetherian, if and only if R[[x]] is Nil∗-Noetherian. Then we discuss some Nil∗-Noetherian properties on idealizations and bi-amalgamated algebras. Finally, we give the Cartan-Eilenberg-Bass Theorem for Nil∗-Noetherian rings in terms of Nil∗-injective modules and Nil∗-FP-injective modules. Besides, some examples are given to distinguish Nil∗-Noetherian rings, Nil∗-coherent rings and so on.