论文标题
nil $ _ {\ ast} $ - noetherian戒指
Nil$_{\ast}$-Noetherian rings
论文作者
论文摘要
在本文中，我们说一个环$ r $是nil $ _ {\ ast} $ - noetherian，前提是任何零零是有限生成的。首先，我们表明希尔伯特基础定理适用于nil $ _ {\ ast} $ - noetherian戒指，也就是说，$ r $是nil $ _ {\ ast} $ - noetherian and noeetherian，并且仅当$ r [x] $ nil $ _ nil $ _ {\ ast ast} $ _ {\ ast} $ - noeetherian，并且只有$ r [x] nil $ _ {\ ast} $ - noetherian。然后，我们讨论一些nil $ _ {\ ast} $ - noetherian属性在理想化和双重组合的代数方面。最后，我们为nil $ _ {\ ast} $ - Noetherian的nil $ _ {\ ast} $ - 注射模块和nil $ _ {\ ast} $ fp indective模块提供了cartan-eilenberg-bass定理。此外，还提供了一些示例来区分nil $ _ {\ ast} $ - noetherian戒指，nil $ _ {\ ast} $ - 相干戒指等。
In this paper, we say a ring $R$ is Nil$_{\ast}$-Noetherian provided that any nil ideal is finitely generated. First, we show that the Hilbert basis theorem holds for Nil$_{\ast}$-Noetherian rings, that is, $R$ is Nil$_{\ast}$-Noetherian if and only if $R[x]$ is Nil$_{\ast}$-Noetherian, if and only if $R[[x]]$ is Nil$_{\ast}$-Noetherian. Then we discuss some Nil$_{\ast}$-Noetherian properties on idealizations and bi-amalgamated algebras. Finally, we give the Cartan-Eilenberg-Bass Theorem for Nil$_{\ast}$-Noetherian rings in terms of Nil$_{\ast}$-injective modules and Nil$_{\ast}$-FP-injective modules. Besides, some examples are given to distinguish Nil$_{\ast}$-Noetherian rings, Nil$_{\ast}$-coherent rings and so on.
