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](https://en.wikipedia.org/wiki/Perfect_ring) to describe perfect rings.)



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



A perfect ring that isn't semiprimary
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.)


