Example of a Non-Graded Ideal in a Graded Ring

Question

A ring $S$ is said to be graded if there are additive subgroups $S_0, S_1, S_2, \ldots$ such that $S=\bigoplus_{k\geq 0}S_k$ and $S_iS_j\subseteq S_{i+j}$ for all $i$ and $j$.

An ideal $I$ in a graded ring $S$ is said to be graded if $I=\bigoplus_{k\geq 0}I\cap S_k$.

Can somebody give an example of a graded ring having an ideal which is not graded?

Thanks.

Answer 1

If $S=K[X,Y]$ with the usual grading, that is, $S_n=\sum_{i+j=n} KX^iY^j$, then the ideal $I=(X+Y^2)$ is not graded (why?).

Even simpler, $S=K[X]$ with $S_n=KX^n$ and $I=(X+1)$.