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.