Yang, Xiaoyan; Liu, Zhongkui Generalized Noetherian property of rings. (English) Zbl 1140.16007 J. Math. Res. Expo. 26, No. 4, 674-678 (2006). A ring \(R\) is called right Ne-Noetherian (resp. right U-Noetherian) if it satisfies the ascending chain condition for non-essential (resp. uniform) right ideals. A right \(R\)-module \(Q\) is called Ne-injective (resp. U-injective) if for each non-essential (resp. uniform) right ideal \(I\) of \(R\), every \(R\)-homomorphism \(f\colon I\to Q\) extends to \(R\). Motivated by the well-known result that a ring \(R\) is right Noetherian if and only if direct sums of injective right \(R\)-modules are injective, the authors prove that \(R\) is right Ne-Noetherian if and only if direct sums of Ne-injective modules are Ne-injective and if unions of countable ascending chains of non-essential right ideals are non-essential right ideals. Furthermore, \(R\) is shown to be right U-Noetherian if and only if direct sums of U-injective modules are U-injective. Reviewer: Günter Krause (Winnipeg) MSC: 16P40 Noetherian rings and modules (associative rings and algebras) 16D50 Injective modules, self-injective associative rings Keywords:Ne-injective modules; Ne-Noetherian rings; U-injective modules; U-Noetherian rings; direct sums PDFBibTeX XMLCite \textit{X. Yang} and \textit{Z. Liu}, J. Math. Res. Expo. 26, No. 4, 674--678 (2006; Zbl 1140.16007)