ac.commutative algebra - Examples of finite local rings of length 2 or 3

I'm just going to consider the local rings with residue field $mathbf{F}_p$.

Suppose A' is an extension of $mathbf{Z}/p$ with ideal $mathbf{Z}/p$. If we take the fiber product with $mathbf{Z}$, we get an extension of $mathbf{Z}$ with ideal $mathbf{Z}/p$. Up to isomorphism, there is only one such extension: $B' = mathbf{Z}[t] / (t^2, pt)$. The kernel of the map from B' to A' is isomorphic to $mathbf{Z}$ and reduces modulo t to the ideal generated by p. Therefore, the square-zero extensions of $mathbf{Z}/p$ are all isomorphic to $mathbf{Z}[t] / (t^2, pt, p + lambda t)$ for some $lambda not= 0$. If $lambda$ is not a multiple of p, we get $mathbf{Z} / p^2$; if $lambda$ is a multiple of $p$, we get $mathbf{Z}[t] / (p, tp, t^2) = mathbf{F}_p[t] / t^2$. So these are all the length 2 finite local rings with residue field $mathbf{F}_p$.

For length 3, we'll look for extensions of $mathbf{Z} / p^2$ by $mathbf{Z} / p$. The same analysis shows that these are all of the form $mathbf{Z}[t] / (t^2, pt, p^2 + lambda t)$. If $lambda$ is not divisible by $p$, we get $mathbf{Z} / p^3$ and if $lambda$ is divisible by $p$ we get $mathbf{Z}[t] / (p^2, pt, t^2)$.

We also have to look for extensions of $mathbf{F}_p[t] / t^2$. By base change, any such extension A' gives an extension B' of $mathbf{Z}[t]$ with ideal $mathbf{Z} / p$. Once again, there is only one of these up to isomorphism (since $mathbf{Z}[t]$ is projective, for example) and it is given by $mathbf{Z}[t,u] / (u^2, pu, tu)$. The ideal of the map from B' to A' generates the ideal of the map from $mathbf{Z}[t]$ to $mathbf{F}_p[t] / t^2$. Since this is generated by p and t^2 the ideal of A' in B' is generated by $(p + lambda u, t^2 + mu u)$ and A' is of the form

$mathbf{Z}[t,u] / (u^2, pu, tu, p + lambda u, t^2 + mu u)$

for some polynomials $lambda, mu in mathbf{Z}[t]$.

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Edit: If $lambda$ is not in $(p, t)$ then it is invertible in the quotient, so we get

$mathbf{Z}[t,u] / (p^2, tp, t^2 + mu lambda^{-1} p)$.

There are two possibilities up to isomorphism here, depending on whether $- mu lambda^{-1}$ is a quadratic residue modulo p.

If $lambda$ is in $(p,t)$ we get

$mathbf{Z}[t,u] / (u^2, pu, tu, p, t^2 + mu u) = mathbf{F}_p[t,u] / (u^2, tu, t^2 + mu u)$.

The $mu$ is also not in $(p, t)$ then we get $mathbf{F}[t] / t^3$. If $mu$ is in $(p,t)$ we get $mathbf{F}_p[t,u] / (u^2, tu, t^2)$.

Modulo any mistake I made above, I think a complete list is of length 3 finite local rings with residue field $mathbf{F}_p$ is

1. $mathbf{Z} / p^3$,


2. $mathbf{Z}[t] / (p^2, pt, t^2)$,


3. $mathbf{Z}[t] / (t^2 - p, t^3)$,


4. $mathbf{Z}[t] / (t^2 - alpha p, t^3)$ where $alpha$ is a non-quadratic residue modulo $p$,


5. $mathbf{F}_p[t] / t^3$, and


6. $mathbf{F}_p[t,u] / (t,u)^2$.