ω~T-Noetherian Rings And μ-Noetherian Rings

In this paper, we introduce gυ-ideal, gυ-torsion-free module,ω~T-module,ω~T-envelope in Commutative rings and defineω~T-Noetherian ring. We discuss as-sociate prime ideals and primary decomposition ofω~T-Noetherian rings. Byintroducingω~T-irreducible ideal, we prove that any properω~T-ideal has primarydecomposition of primaryω~T-ideal. By studying the structure and propertiesof injective modules ofω~T-Noetherian rings, we extend Cartan-Eilenberg-BassTheorem toω~T-Noetherian rings: R is aω~T-Noetherian ring if and only if everydirect sum of gv -torsion-free injective modules is injective; if and only if everygv -torsion- free injective module isΣ-injective. Let R be aω~T-Noetherian ring,we also prove that E(R/I) can be presented finite direct sum of indecomposableinjective modules if I is a properω~T-ideal. Finally, we defineυ-Noetherian ring(ie. having the condition of ascending chain onυ-ideals). We prove that everyω~T-Noetherian ring is aυ-Noetherian ring. Let R be aυ-Noetherian ring, weprove that R[P] is aυ-Noetherian ring if P is a prime ideal. We also prove that ifeach nonzero ideal is contained in at most finitely many maximal t-ideals and foreach maximal t-ideal M, R[M] is aυ-Noetherian ring, then R is aυ-Noetherianring.