a problem involving a homogeneous ideal and an infinite field (Matsumura, CRT, 13.1) I am trying to solve the following problem (this is 13.1 from Matsumura's Commutative Ring Theory):
Prove the following: (i) Let $R= \bigoplus_{n\ge0}R_n$ be a graded ring. Then for any $u \in R_0^*$ the map $T_u(\sum x_n) = \sum x_n u^n$ is an automorphism of $R$. (ii) if $R_0$ contains an infinite field $k$ then an ideal $I$ of $R$ is homogeneous if and only if $T_a(I)=I, \, \forall a \in k$.
It is not hard to prove (i) and also the $\Rightarrow$ direction of $(ii)$. But i am stuck in proving that if $T_a(I)=I, \, \forall a \in k$, then $I$ is homogeneous. Here are some thoughts: I am trying to see what observations i can make. 1) every element $a$ of $k$ induces an automorphism of $R$. 2) for every $a,b \in k$ we have $T_a(I)=T_b(I)=I$. 3) every homogenous component of $I$, that is $I \cap R_n$ is a $k$-vector space. 4) A vector space over an infinite field can not be the union of proper subspaces. 
That's all i can think of and i seem to be unable to exploit any of these.
I would appreciate the smallest of hints to get me going with this problem, cause i feel that whatever the key is, i am completely missing it.

Hint: If $v_i$ are homogeneous and $x = v_0 + v_1 + \cdots + v_n$ is an element of $I$ then for any nonzero $a \in k$ the element $y = T_a(x) - a^nx$ is an element of $I$.  You'll need to use the fact that $k$ is infinite to choose $a$ so that $y \neq 0$.
