Uniformly S-Noetherian rings

Abstract

Let R be a ring and S be a multiplicative subset of R. Then R is called a uniformly S-Noetherian ring if there exists s ∈ S such that, for any ideal I of R, sI ⊆ K for some finitely generated subideal K of I. We give the Eakin-Nagata-Formanek theorem for uniformly S-Noetherian rings. In addition, the uniformly S-Noetherian properties on several ring constructions are given. The notion of u-S-injective modules is also introduced and studied. Finally, we obtain the Bass-Papp theorem for uniformly S-Noetherian rings.

Mathematics Subject Classification (2020):

• 13E05
• 13A15
• 13C11
• 13D30

Key words:

• Uniformly S-Noetherian ring
• uniformly S-Noetherian module
• u-S-injective module
• trivial extension
• pullback
• amalgamated algebra