Normal ideals of graded rings

For a graded domain $R=k[X_0,\mathrm{.}\mathrm{.}\mathrm{.},X_m]\mathrm{/}J$ over an arbitrary domain $k$, it is shown that the ideal generated by elements of degree $\ge mA$, where $A$ is the least common multiple of the weights of the $X_i$, is a normal ideal.