On Near Armendariz Ideals

We say a left ideal I is a near Armendariz whenever polynomials (x)f = f(0) + f(1)x + ... + f(m)x(m), (x)g = g(0) + g(1)x + ... + g(n)x(n) is an element of R[x] satisfy (x)fo(x)g is an element of r(R[x]) (I[x]) then g(j)f(i) is an element of r(R)(I) for each 1 <= i <= m, 1 <= j <= n and (f(0))g is an element of r(R)(I). The behavior of the left near Armendariz ideal condition is 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.