Equivalent properties of graded ideals 
Let $S=k[x_1,\ldots,x_n]$ be a polynomial ring over a field $k$. Note that $S=\oplus_{i\in\mathbb{N}}S_i$ where $S_i$ is the $k$-space spanned by all monomials of total degree $i$. Therefore any $f\in S$ has a unique representation $f=\sum f_i$ where $f_i\in S_i$. Call the $f_i$ the homogeneous component of $f$ of degree $i$. Let $J\subset S$ be an ideal. We say $J$ is graded or homogeneous if it satisfies one of the following equivalent properties:

*

*If $f\in J$ then every homogeneous component of $f$ is in $J$.

*$J=\oplus_{i\in\mathbb{N}} J_i$ where $J_i=S_i\cap J$.

*If $\tilde{J}$ is the ideal generated by all homogeneous elements of $J$, then $\tilde{J}=J$.

*$J$ has a set of homogeneous generators.


I want to prove that properties 1-4 are equivalent.
Here's what I have so far:
$(1\implies 2)$ Let $f\in J$ and write $f$ as the sum of its homogeneous components $f=\sum f_i$, $f_i\in S_i$. By assumption, each $f_i\in J$. Thus $f\in\sum_{i\in\mathbb{N}} J_i$. Since $S$ is the direct sum of the $S_i$, this representation of $f$ is unique, hence we have $f\in\oplus_{i\in\mathbb{N}} J_i$. The other inclusion is clear, so we have equality.
$(2\implies 3)$ Clearly $\tilde{J}\subset J$. Let $f\in J$. Then by assumption, we have $f=\sum f_i$ where each $f_i\in J_i=S_i\cap J$. Thus each $f_i$ is a homogeneous element of $J$, so $f\in\tilde{J}$. Hence equality holds.
$(3\implies 4)$ I think this is trivial.
$(4\implies 1)$ Here is where I get stuck. If we let $\{g_i\}$ be a set of homogeneous generators of $J$, then if $f\in J$ we can write $f=\sum\alpha_ig_i$ where each $\alpha_i\in S$. How can I show every homogeneous component of $f$ lives in $J$? Something to do with distributing the $f_i$ into the $\alpha_i$ and working from there?
Also, are my other implications fine?
 A: Let $f\in J$ and $f_n$ denote the  degree-$n$ homogeneous component of $f$. We have to show that $f_n\in J$.
Let $\{g_\ell\}$ be a set of homogeneous generators of $J$ and $s_\ell=\deg g_\ell$. We can write $$f=a_1g_1+\cdots+a_r g_r$$ for some $g_1,\cdots, g_r\in \{g_\ell\}$ and $a_1,\cdots,a_r\in S$. Write $a_\ell =\sum_k a_{\ell, k}$ where $a_{\ell, k}$ is the degree-$k$ homogeneous component of $a_\ell$. Thus we have $$f_n=a_{1,n-s_1}g_1+\cdots+a_{r,n-s_r}g_r.$$ Since $\{g_\ell\}\subseteq J$, we conclude that $f_n\in J$.
A: I think I got it figured out.
Let $\{g_\ell\}$ be a set of homogeneous generators of $J$. Let $f\in J$. So we have
$$
f=\sum \alpha_\ell g_\ell,\;\;g_\ell\in S_\ell\;\text{ and }\;\alpha_\ell\in S.
$$
Writing
$$
\alpha_\ell=\sum a_{\ell,m}\;\;\text{where}\;a_{\ell,m}\in S_m,
$$
we obtain
$$
f=\sum\sum a_{\ell,m}g_\ell\tag{1}
$$
where each $a_{\ell,m}g_\ell$ is homogeneous of degree $\ell+m$. We have essentially just distributed the $g_\ell$ across the $\alpha_\ell$. From equation $(1)$ it is now clear that the homogeneous components of $f$ are just $S$-combinations of the $g_\ell$, and therefore belong to $J$.
Please point out any mistakes.
