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In 2011, Howlett and Nguyen \cite{r1} introduced the concept of a $W$-graphideal $E_J$ in $\left ( W,\leqslant_{L} \right )$ with respect to $J$ (a subsetof $S$), where $\leqslant _{L}$ is the left weak order on $W$. They proved thatone can construct a $W$-graph from a given $W$-graph ideal by constructing aHecke module structure on $E_J$, where the $W$-graph was introduced by Kazhdanand Lusztig in \cite{d1}.  In this paper, we give the relation between Hecke modules on $E_J$ andgeneral Hecke algebras by considering the relation between Hecke modules on$E_J$ and parabolic Hecke modules. And inspired by Lusztig \cite{g3}, we showthat the parabolic Hecke module is isomorphic to a left ideal of the Heckealgebra. Lastly, we give the relation between $R$-polynomials on $E_J$ andparabolic $R$-polynomials as an application of the main results.

The relation between Parabolic Hecke modules and $W$-graph ideal modules in Kazhdan-Lusztig theory