---
tag: VW2H7873YM
kind: statement
role: theorem
name: Amitsur's theorem on the radical of polynomial rings
msc: [16N20, 16S36]
uses: [0894J980RK, 0Q9EZS90GY, XA0DCNH7WY, NZ1G9WRWFB, 019NFGFPR5]
about: [XA0DCNH7WY, 0SDC934R52]
sources:
  - url: https://en.wikipedia.org/wiki/Jacobson_radical
---

## Statement

Let $R$ be a [[ring|0894J980RK]] and let $R[x]$ be the [[polynomial ring|0Q9EZS90GY]] in a central variable $x$ over $R$.
Let $J(R[x])$ be the [[Jacobson radical|XA0DCNH7WY]] of $R[x]$ and let $N = J(R[x]) \cap R$.
Then $N$ is a [[nil ideal|NZ1G9WRWFB]] of $R$ and $J(R[x]) = N[x]$, the set of polynomials with all coefficients in $N$.

## Remarks

The harder part is to show that $J(R[x])$ is generated by its constant polynomials.
The nil part is then direct: for $a \in N$, the element $ax$ lies in $J(R[x])$, so $1 - ax$ is a [[unit|019NFGFPR5]] of $R[x]$.
Its inverse must be the power series $\sum_{n \ge 0} a^n x^n$, which is a polynomial only if $a^n = 0$ for large $n$.
Example: for a commutative ring $R$, $J(R[x])$ is $\mathfrak{N}[x]$, where $\mathfrak{N}$ is the [[nilradical|QQDN6EYQNJ]] of $R$; for instance $J(\Z[x]) = 0$, although $\Z$ has many maximal ideals.
In particular, if $R$ has no nonzero nil ideals, then $R[x]$ is [[semiprimitive|1B2DBKV1C9]].

## Explanations

### L1 human

The Jacobson radical of a ring of polynomials is always made of polynomials whose coefficients are nilpotent.
So adjoining a variable kills most of the radical: only the nilpotent part survives.

### L2 human

The theorem is a standard tool for producing semiprimitive rings, since $R[x]$ is semiprimitive whenever $R$ has no nonzero nil ideals.
It also connects the Jacobson radical with nil ideals and with the [[Köthe conjecture|RT2D9JEVRR]], which in Krempa's form asks whether $J(R[x]) = R[x]$ for every nil ring $R$.