In a commutative algebra, every nil ideal is nilpotent, right?

Question

Can anyone prove or disprove the following: In a commutative algebra, every nil ideal is nilpotent.

The reason I am asking is that I have an ideal which I can show to be nil ideal but which might not be a nilpotent ideal.

Answer 1

Nope, this is false: in $R=k[x_1,x_2,\dots]$ for some field $k$, consider the ideal $I=\langle x_1,x_2^2,x_3^3,\dots\rangle$. Then $$\langle x_1,x_2,x_3,\dots\rangle\lhd R/I$$ is nil but not nilpotent.

However, if your ring (or algebra in your case) is Noetherian, nil ideals are nilpotent. This is known as Levitzky's theorem. A good reference I can give you on this is Lam's book A First Course in Noncommutative Rings.