On the extent of the injectivity of direct sums of modules

Abstract

Inspired by the fact that a ring is right Noetherian if and only if direct sums of certain injective right modules are once again injective, we propose a way to not only decide whether a ring is Noetherian but to gauge how Noetherian an arbitrary ring is. As an application of this line of thinking, volatile rings, a notion opposite to Noetherianness, are introduced. Characterizations for right volatile rings, meant as counterparts to corresponding characterizations of right Noetherian rings, are given. Examples of volatile rings as well as of rings that are neither Noetherian nor volatile are presented.

Mathematics Subject Classification (2020):

• Primary: 16P40
• 16D50
• Secondary: 16S90

Key words:

• Noetherian rings
• injective modules
• relative injectivity
• direct sums