Standalone Proof for Hilbert's Nullstellensatz Theorem

Hilbert's Nullstellensatz Theorem has been one of the most influential building block of algebraic geometry. The theorem provides a dictionary connecting algebra and geometry by stating that any polynomial that vanishes on the common zeros of a set of polynomials must belong to the radical of the ideal generated by those polynomials. Try coming up with a self-sustaining proof for weak and strong Nullstellensatz from first principle that allows anyone with basic understanding of Ring theory to understand the proof.