A perfect ring that isn't semiprimary

Question

It's well-known that a semiprimary ring (meaning $R/J(R)$ is semisimple and $J(R)$ is nilpotent) is left and right perfect (there are a lot of ways to describe perfect rings.)

What's a good (hopefully commutative) example of a left and right perfect ring that isn't semiprimary?

I'm aware of one-sided perfect rings with non-nilpotent Jacobson radicals, but I've had some trouble digging up a two-sided perfect ring with a non-nilpotent Jacobson radical. I don't think I've ever run across any literature to the effect that there aren't such examples.

If I knew how to give an example of a commutative $T$-nilpotent ring that is a module over a field and is not nilpotent, then I could just use the Dorroh extension of that.

Answer 1

Let $k$ be a field. Consider the commutative algebra of polynomials $A=k[x_1,x_2,x_3,\dotsc]$ in a countable set of variables $x_1$, $x_2$, $x_3,\,\dots$ with coefficients in $k$. Let $I$ be the ideal in $A$ generated by the following elements:

$x_n^2$, for every $n\ge1$;

$x_ix_j$, for every $i\ge1$ and $j\ge2i$.

Then the quotient algebra $B=A/I$ is a commutative local $k$-algebra whose maximal ideal $J$ (generated by the elements $x_1$, $x_2$, $x_3,\,\dots$) is $T$-nilpotent, but not nilpotent.

Indeed, every monomial in the variables $x_1$, $x_2$, $x_3,\,\dots$ divisible by $x_n$ and having length exceeding $n$ for some $n\ge1$ vanishes in $B$. So the ideal $J\subset B$ is $T$-nilpotent and the ring $B$ is perfect.

But the monomial $x_nx_{n+1}\dotsm x_{2n-1}$ of length $n$ is nonzero in $B$ for every $n\ge1$. So the Jacobson radical (= nilradical) $J$ of the ring $B$ is not nilpotent.

(I haven't invented this example for the occasion, but rather have recently seen it somewhere. But I forgot the source.)