The Armendariz module and its application to the Ikeda-Nakayama module

Abstract

A ring R is called a right Ikeda-Nakayama (for short IN-ring) if the left annihilator of the intersection of any two right ideals is the sum of the left annihilators, that is, if ℓ(I∩J)=ℓ(I)+ℓ(J) for all right ideals I and J of R. R is called Armendariz ring if whenever polynomials f(x)=a0+a1x+⋯+amxm, g(x)=b0+b1x+⋯+bnxn∈R[x] satisfy f(x)g(x)=0, then aibj=0 for each i,j. In this paper, we show that if R[x] is a right IN-ring, then R is a right IN-ring in case R is an Armendariz ring.