## § Radical ideals, nilpotents, and reduced rings

#### § Radical Ideals

A radical ideal of a ring $R$ is an ideal such that
$\forall r \in R, r^n \in I \implies r \in I$.
That is, if any power of $r$ is in $I$, then the element
$r$ also gets "sucked into" $I$.
#### § Nilpotent elements

A nilpotent element of a ring $R$ is any element $r$ such that there exists
some power $n$ such that $r^n = 0$.
Note that every ideal of the ring contains $0$. Hence, if an ideal $I$
of a ring is known to be a radical ideal, then for any nilpotent $r$,
since $\exists n, r^n = 0 \in I$, since $I$ is radical, $r \in I$.
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 $I$, it's radical idea $\sqrt I \equiv \{ r \in R, r^n \in I \}$.
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, $\sqrt{(0)} \equiv \{ r \in R, r^n = 0\}$.
#### § Reduced rings

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 $R' \equiv R / \sqrt{(0}$. Since
$\sqrt{(0)}$ 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.
A ring modulo a radical ideal is reduced

#### § Integral domains

a Ring $R$ is an integral domain if $ab = 0 \implies a = 0 \lor b = 0$. That is,
the ring $R$ has no zero divisors.
#### § Prime ideals

An ideal $I$ of a ring $R$ is a prime ideal if
$\forall xy \in R, xy \in I \implies x \in I \lor y \in I$. This generalizes
the notion of a prime number diving a composite: $p | xy \implies p | x \lor p | y$.
#### § creating integral domains by quotenting prime ideals

Recall that every ideal contains a $0$. Now, if an ideal $I$ is prime, and if
$ab = 0 \in I$, then either $a \in I$ or $b \in I$ (by the definition of prime).
We create $R' = R / I$. We denote $\overline{r} \in R'$ as the image of $r \in R$
in the quotient ring $R'$.
The intuition is that quotienting by a $I$, since if $ab = 0 \implies a \in I \lor b \in I$,
we are "forcing" that in the quotient ring $R'$, if $\overline{a} \overline{b} = 0$, then either
$\overline{a} = 0$ or $\overline{b} = 0$, since $(a \in I \implies \overline a = 0)$,
and $b \in I \implies \overline b = 0)$.
A ring modulo a prime ideal is an integral domain.

I learnt of this explanation from this
excellent blog post by Stefano Ottolenghi .