Amitsur's theorem on the radical of polynomial rings

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 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 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.

Going deeper

For a student

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, which in Krempa's form asks whether $J(R[x]) = R[x]$ for every nil ring $R$.