The Dictionary

For every ring $A$, we can define the spectrum of $A$, which is but the prime ideals of the ring. So, any prime ideal in the ring corresponds to a point in the affine scheme. Any element $x$ in $A$ can be written as a function in Spec $A$. The radical ideals of $A$ are the closed subsets of Spec $A$. The maximal ideals of $A$ are the closed points of Spec $A$. An affine scheme is a ringed space that is isomorphic to $(Spec A, \Theta_{Spec A})$, which we have written for a ring $A$ where $\Theta_{Spec A}$ is the Zariski topology defined on the set. Read a similar post.

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A radical of an ideal $I \subset A$ is defined as a set 

$$\rm{rad}(I) = \{  r \in A | r^n \in I\ \forall n \in Z^+\}$$

and a nilradical would be a radical of a zero ideal

$$\rm{nilrad}(I) = \{  r \in A | r^n=0 \in I\ \forall n \in Z^+\}$$