A nilpotent element of a ring R is any element r such that there exists some power n such that rn=0. Note that every ideal of the ring contains 0. Hence, if an ideal Iof a ring is known to be a radical ideal, then for any nilpotent r, since ∃n,rn=0∈I, since I is radical, r∈I. That is, a radical ideal with always contain all nilpotents! It will contain other elements as well, but it will contain nilpotents for sure.
Given a ideal I, it's radical idea I≡{r∈R,rn∈I}. That is, we add all the elements I needs to have for it to become a radical. Notice that the radicalization of the zero ideal I will precisely contain all nilpotents. that is, (0)≡{r∈R,rn=0}.
A ring R is a reduced ring if the only nilpotent in the ring is 0.
§ creating reduced rings (removing nilpotents) by quotienting radical ideals
Tto remove nilpotents of the ring R, we can create R′≡R/(0. Since (0) is the ideal which contains all nilpotents, the quotient ring R′ will contain no nilpotents other than 0. Similarly, quotienting by any larger radical ideal I will remove all nilpotents (and then some), leaving a reduced ring.
An ideal I of a ring R is a prime ideal if ∀xy∈R,xy∈I⟹x∈I∨y∈I. This generalizes the notion of a prime number diving a composite: p∣xy⟹p∣x∨p∣y.
§ creating integral domains by quotenting prime ideals
Recall that every ideal contains a 0. Now, if an ideal I is prime, and if ab=0∈I, then either a∈I or b∈I (by the definition of prime). We create R′=R/I. We denote r∈R′ as the image of r∈Rin the quotient ring R′. The intuition is that quotienting by a I, since if ab=0⟹a∈I∨b∈I, we are "forcing" that in the quotient ring R′, if ab=0, then either a=0 or b=0, since (a∈I⟹a=0), and b∈I⟹b=0).
A ring modulo a prime ideal is an integral domain.