I need to show that an ideal $I$ of a $\mathbb{Z}$-graded ring R is homogeneous iff for every element $f \in I$, all homogeneous components of $f$ are in $I$.
$\Leftarrow$ implication is obviously. $\Rightarrow$ implication is also obvious because if $I=(\{g_i\}) \ni f = \sum m_ig_i \space(g_i \in R_j$ for some $j$) and homogeneous components of $f$ are also combination of $g_i$.
But looking at the other proofs(such this) i realized i made a mistake, but i do not know where.
P.S. $m_i^{(j)}$ is the homogeneous component of $m_{i}$ with $\deg m_i^{(j)} = j$. Similarly for $f_i$. $f_i = \sum\limits_{j:j + \deg(g_j) = i,r:any}m_{r}^{j}g_j \implies f_i \in I$