On Near Armendariz Ideals

Khadijeh Khalilnezhad, Hamid Haj Seyyed Javadi

Abstract

We say a left ideal $I$ is a near Armendariz whenever polynomials‎

‎$(x) f = f_0‎ + ‎f_1 x‎ + ‎\ldots‎ + ‎f_m x^m$‎, ‎$(x) g = g_0‎ + ‎g_1 x‎ + ‎\ldots‎ + ‎g_n x^n \in R [x]$ satisfy $(x) f o (x)g \in r_{R[x]}(I[x])$ then $ g_j f_i\in r_{R}(I)$ for each $1\leq i \leq m$‎, ‎$1\leq j \leq n$ and $(f_0)g\in r_{R}(I)$‎.

‎The behavior of the left near Armendariz ideal condition is\linebreak investigated with respect to various constructions‎. ‎It is shown that every left ideal of a near Armendariz ring is a near Armendariz left ideal‎. ‎Examples are included to illustrate and delimit the theory‎.