Help with an implication of a really basic question in algebraic geometry I'm starting to study algebraic geometry and I'm trying to prove this implication:

For a homogeneous ideal $\mathfrak a\subset S$, show that $Z(\mathfrak
 a)=\emptyset \implies \sqrt {\mathfrak a}=\text{either}\ S \ \text{or
 the ideal}\ S_+=\bigoplus_{d\gt0}S_d$.

I have seeing somewhere that this is true because of these implications:
$Z(\mathfrak a)$ is empty $\implies$ in $\mathbb A^{n+1}$, $Z(\mathfrak a)$ must be empty or $(0,\ldots 0) \implies \sqrt{\mathfrak a}=S$ or $\sqrt{\mathfrak a}=\bigoplus_{d\gt 0} S_d$.
I don't understand both implications, I would appreciate if someone could help me, I really need help.
Thanks.
 A: I assume that $S=k[x_0,...,x_n]$, where $k$ is an algebraically closed field. This is exactly your setup, no?
Anyways, the homogeneous ideal $\mathfrak{a} \subset S$ defines a projective algebraic set $Z^{proj}(\mathfrak{a}) \subset \Bbb P^n$. But $\mathfrak{a}$ is also an honest ideal of the polynomial ring $k[x_0,...,x_n]$, hence it also defines an affine algebraic set
$$
Z^{aff}(\mathfrak{a}) \subset \Bbb A^{n+1}
$$
Note that by construction $\Bbb P^n$ is a quotient of $\Bbb A^{n+1} \setminus \{0\}$. Denote by $[P] \in \Bbb P^n$ the residue class of $P \in \Bbb A^{n+1} \setminus \{0\}$.
Now if $P \in Z^{aff}(\mathfrak{a})$ and $P \neq 0$, then $[P] \in Z^{proj}(\mathfrak{a})$, so if $Z^{proj}(\mathfrak{a})$ is empty then either $Z^{aff}(\mathfrak{a})$ is empty as well, or $Z^{aff}(\mathfrak{a})=\{0\}$.
The second implication follows from the well-known order preserving bijection between radical ideals and affine algebraic sets over an algebraically closed field, which in turn follows from Hilbert's Nullstellensatz. In particular, if $Z^{aff}(\mathfrak{a})=\emptyset$ then
$$
\sqrt{\mathfrak{a}}=\mathfrak{a}=S
$$
and if $Z^{aff}(\mathfrak{a})=\{0\}$ then
$$
\sqrt{\mathfrak{a}}=(x_0,...,x_n)=\bigoplus_{d>0}S_d
$$
