×

Generalized Noetherian property of rings. (English) Zbl 1140.16007

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.

MSC:

16P40 Noetherian rings and modules (associative rings and algebras)
16D50 Injective modules, self-injective associative rings
PDFBibTeX XMLCite