NPP RINGS, REDUCED RINGS AND SNF RINGS

Authors : Junchao WEİ, Jianhua CHEN

Pages : 9-26

View : 13 | Download : 7

Publication Date : 2008-12-01

Article Type : Research Paper

Abstract :A ring R is called left NP P if for any nilpotent element a of R, linsert ignore into journalissuearticles values(a); = Re, e2 = e ∈ R. A right R−module M is called N f lat if for each a ∈ Ninsert ignore into journalissuearticles values(R);, the Z−module map 1M ⊗ i : M ⊗R Ra −→ M ⊗R R is monic, where i : Ra ,→ R is the inclusion map. A ring R is called right SNF if every simple right R−module is N f lat. In this paper, we first show that a ring R is left NP P iff every sum of two injective submodules of a left R−module is nil−injective. And some properties of left NP P rings are given, for example, if R is left NP P, so is eRe for any e2 = e ∈ R satisfying ReR = R. Next, we study some properties of reduced rings. A ring R is reduced if and only if R is ZC and right SNF if and only if R is left and right NP P and R has no subrings which is isomorphic to the upper triangular matrix UTZ2 or UTinsert ignore into journalissuearticles values(Zp);2 for some prime p. Finally, we give some characterizations of n−regular rings, for example, a ring R is n−regular if and only if every right R−module is N flat.
Keywords : nil injective modules, NP P rings, reduced rings, NC2 rings, n regular rings, SNF rings, N f lat modules