What is the intuition (hopefully, geometric) behind these basic facts about homogeneous ideals? An ideal $I$ in $S$ is homogeneous if an element $f = \sum_{n \ge 0} f_n$ of $S$ lies in $I$ if and only if each homogeneous part $f_n$ lies in $I$. Here, we let $S = \bigoplus_{n \ge 0} S_n$ be an $\mathbf{N}$-graded ring.

An ideal generated by homogeneous elements is a homogeneous ideal.

A homogeneous ideal $I$ in $S$ is prime if and only if it is a proper ideal and $$fg \in I \implies f \in I \text{ or }g \in I$$for homogeneous $f, g \in S$.

The kernel of $\mathbf{N}$-graded rings is a homogeneous ideal.

For any homogeneous ideal $I$ of $S$ there is a natural $\mathbf{N}$-grading on $S/I$.

I am not an algebraic geometrer by trade, but I need to use these results.