Search

Journal

This page serves as a rolling journal of thoughts and ideas from my daily flights of mathematical lunacy. By no means am I an expert in any of these fields. This will be a more informal and casual approach to some advanced mathematical topics in my own words. Yes there will be mistakes here and there, this is not peer reviewed or nearly as rigorous as a research level report. So take everything with a grain of salt. However, I hope this encourages you to learn something new about an interesting field of math or maybe learn about fields of math you never knew existed. Always feel free to reach out if you have questions or thoughts, I love discussing mathematics and I love learning from others. Cheers.

Latest entry:

Radicals (of Rings)

2026-03-25

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. Ideals are the ring theory analogue of subgroups.

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. The nilradical of a ring $R$ is the set of all nilpotent elements, that is $$ \operatorname{Nil}(R) = \{x \in R: x^n = 0 \text{ for some } n \ge 1\} $$ Further, it is the intersection of all the prime ideals of $R$. $$ \operatorname{Nil}(R) = \bigcap_{\mathfrak{p} \in R} \mathfrak{p} $$ So we can think of the nilradical as identifying all the elements that "vanish" on prime ideals. Take for example $\mathbb{Z}_{12}$. We have $6^2 = 36 \equiv 0 \mod 12$. So $6$ is a nilpotent element. So the nilradical is $\operatorname{Nil}(\mathbb{Z}_{12}) = \{0,6\}$.

Of course, we have a similar notion of a radical in the context of maximal ideals, namely, the Jacobson radical. For a ring $R$, the Jacobson radical is the intersection of all the maximal ideals of $R$, that is $$ J(R) = \bigcap_{\mathfrak{m} \in R} \mathfrak{m} $$ A useful equivalent definition is that an element $x$ is in $J(R)$ iff $1-rx$ is a unit (invertible) for all $r \in R$. The Jacobson radical gives a notion of elements that vanish at all points. This is a stronger statement than the nilradical but in the other direction. The big takewawy of today is the inclusion $$ \operatorname{Nil}(R) \subseteq J(R) $$

In summary, think of the nilradical as all nilpotent elements and the Jacobson radical of all elements that kills units via $1-rx$. I'm told that radicals are super important and I think I am just starting to see why. Of course I am pretty new to commutative algebra but it has been really cool to play with radicals and some of their properties. Some of the properties I mentioned today are actually pretty easy to prove so I urge you to try it out and work some more examples in polynomial rings. Cheers.

Permalink →