§ 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 .