NPP RINGS, REDUCED RINGS AND SNF RINGS

Abstract

A ring R is called left NP P if for any nilpotent element a of R, l(a) = Re, e2 = e ∈ R. A right R−module M is called N f lat if for each a ∈ N(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 UT(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