§ Radical ideals, nilpotents, and reduced rings

§ Radical Ideals

A radical ideal of a ring RR is an ideal such that ∀r∈R,rn∈I  ⟹  r∈I\forall r \in R, r^n \in I \implies r \in I. That is, if any power of rr is in II, then the element rr also gets "sucked into" II.

§ Nilpotent elements

A nilpotent element of a ring RR is any element rr such that there exists some power nn such that rn=0r^n = 0. Note that every ideal of the ring contains 00. Hence, if an ideal II of a ring is known to be a radical ideal, then for any nilpotent rr, since ∃n,rn=0∈I\exists n, r^n = 0 \in I, since II is radical, r∈Ir \in 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.

§ Radicalization of an ideal

Given a ideal II, it's radical idea I≡{r∈R,rn∈I}\sqrt I \equiv \{ r \in R, r^n \in I \}. That is, we add all the elements II needs to have for it to become a radical. Notice that the radicalization of the zero ideal II will precisely contain all nilpotents. that is, (0)≡{r∈R,rn=0}\sqrt{(0)} \equiv \{ r \in R, r^n = 0\}.

§ Reduced rings

A ring RR is a reduced ring if the only nilpotent in the ring is 00.

§ creating reduced rings (removing nilpotents) by quotienting radical ideals

Tto remove nilpotents of the ring RR, we can create R′≡R/(0R' \equiv R / \sqrt{(0}. Since (0)\sqrt{(0)} is the ideal which contains all nilpotents, the quotient ring R′R' will contain no nilpotents other than 00. Similarly, quotienting by any larger radical ideal II will remove all nilpotents (and then some), leaving a reduced ring.
A ring modulo a radical ideal is reduced

§ Integral domains

a Ring RR is an integral domain if ab=0  ⟹  a=0∨b=0ab = 0 \implies a = 0 \lor b = 0. That is, the ring RR has no zero divisors.

§ Prime ideals

An ideal II of a ring RR is a prime ideal if ∀xy∈R,xy∈I  ⟹  x∈I∨y∈I\forall xy \in R, xy \in I \implies x \in I \lor y \in I. This generalizes the notion of a prime number diving a composite: p∣xy  ⟹  p∣x∨p∣yp | xy \implies p | x \lor p | y.

§ creating integral domains by quotenting prime ideals

Recall that every ideal contains a 00. Now, if an ideal II is prime, and if ab=0∈Iab = 0 \in I, then either a∈Ia \in I or b∈Ib \in I (by the definition of prime). We create R′=R/IR' = R / I. We denote r‾∈R′\overline{r} \in R' as the image of r∈Rr \in R in the quotient ring R′R'. The intuition is that quotienting by a II, since if ab=0  ⟹  a∈I∨b∈Iab = 0 \implies a \in I \lor b \in I, we are "forcing" that in the quotient ring R′R', if a‾b‾=0\overline{a} \overline{b} = 0, then either a‾=0\overline{a} = 0 or b‾=0\overline{b} = 0, since (a∈I  ⟹  a‾=0)(a \in I \implies \overline a = 0), and b∈I  ⟹  b‾=0)b \in I \implies \overline b = 0).
A ring modulo a prime ideal is an integral domain.
I learnt of this explanation from this excellent blog post by Stefano Ottolenghi .