Radicals of rings are a huge topic of interest in commutative algebra and further, algebraic geometry, operator algebras, and many other areas. Today I just want to look at the basics of the nilradical and the Jacobson radical. We first need to start with prime and maximal ideals.

Given a commutative ring $R$, an ideal $I$ is defined as an additive subgroup of $R$ such that for all $a \in R$ and $b\in I$, we have $ab \in I$. So if we are in a polynomial or function ring, if a function vanishes on an ideal then multiplying it by another function outside the ideal still yields a vanishing function. So in this way, ideals have a notion of closure.

Of course, there are many kinds of ideals but the two we are most interested in are prime and maximal ideals. We say an ideal $\mathfrak{p} \subseteq R$ is prime if for every $ab \in \mathfrak{p}$, it is implied that either $a \in \mathfrak{p}$ or $b \in \mathfrak{p}$. For those familiar with domains, an ideal $\mathfrak{p}$ is prime if $R/\mathfrak{p}$ is an integral domain. In a loose sense, prime ideals detect when products vanish, this is important when looking at factorizations. Next, we have maximal ideals. These are ideals $\mathfrak{m} \subset R$ that are not contained in any other ideals. That is, there exists no ideal $\mathfrak{a}$ such that $\mathfrak{m} \subset \mathfrak{a} \subset R$. These are simple definitions of course but the theory and structure of these particular ideals runs very deep.

Now we have all we need to talk about the radicals of interest. First, we talk about the nilradical of a ring.