THE ARMENDARIZ MODULE AND ITS APPLICATION TO THE IKEDA-NAKAYAMA MODULE

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 ) = a 0 + a 1 x + ⋯ + a m x m , g ( x ) = b 0 + b 1 x + ⋯ + b n x n ∈ R [ x ] satisfy f ( x ) g ( x ) = 0 , then a i b j = 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.