Narrow Results

In this paper, we study the annihilating properties of ideals generated by coefficients of polynomials and power series which satisfy a structural equation. We first show that if $f(x)Rg(x)=0$ for polynomials $f(x)={\sum }_{i=0}^m{a}_i{x}^i,g(x)={\sum }_{j=0}^m{b}_j{x}^j$ over any ring $R$, then for any $i,j$, there exist positive integers $s(i,j)$ and $t(i,j)$ such that $a_{i_1}R{a}_{i_2}R\cdots a_{i_{s(i,j)}}R{b}_j=0$ and $a_{i}R{b}_{j_1}R{b}_{j_2}R\cdots R{b}_{j_{t(i,j)}}=0$, whenever $i_1,i_2,\dots ,i_{s(i,j)}\le i$ and $j_1,j_2,\dots ,j_{t(i,j)}\le j$. Next we prove that if $f(x)Rg(x)=0$ for power series $f(x)={\sum }_{i=0}^{\infty }{a}_i{x}^i,g(x)={\sum }_{j=0}^{\infty }{b}_j{x}^j$ over any ring $R$, then for any $i,j$, there exist positive integers $s(i,j)$ and $t(i,j)$ such that $a_{i_{s(i,j)}}R\cdots R{a}_{i_2}R{a}_{i_1}R{b}_{j_1}R{b}_{j_2}R\cdots R{b}_{j_{t(i,j)}}=0$ when ${\sum }_{p=1}^{s(i,j)}{i}_p+{\sum }_{q=1}^{t(i,j)}{j}_q<(s(i,j)+1)(t(i,j)+1)(i+1)(j+1)$ and $i_p\le i$, $j_q\le j$ for each $p,q$.