Relation between the spectrum of a ring and affine varieties.

Question

When we define the spectrum of a ring, we associate a topology to this space, and we call this topology, the Zariski topology. However, the Zariski topology is the one associated to the space $k^n$ where $k$ is an algebraically closed field, and the closed sets are affine varieties which are the image of the radical ideals of the ring of polynomials. My question is: What is the relation between these two concepts? Is the spectrum of a ring a generalization of affine varieties? Please if you can help me with this, I'll really appreciate it.

Answer 1

Yes, the two concepts are very much related. In fact, they are almost the same. The points of $k^n$ are in one to one correspondence with maximal ideals in the ring $k[x_1,\ldots,x_n]$. One can think of it as all the closed points in $\mathrm{spec}(k[x_1,\ldots,x_n])$. The Zariski topology on $k^n$ is indeed the induced one from the Zariski topology on $\mathrm{spec}(k[x_1,\ldots,x_n])$.