Rings of order $p^2$ and some related information

Question

Although group of order $p^2$ are well known, the rings of order $p^2$ may not be so well known; I was feeling that there could be more than two rings of order $p^2$. I have two questions related to this, and I don't know whether the questions I am posing are trivial. $\mathbb{Z}_{p^2}$ and $\mathbb{Z}_p\oplus\mathbb{Z}_p$ are obvious examples of rings (with unity) of order $p^2$.

Question 1. Are there more than two rings (with unity) of order $p^2$?

Question 2. If there are more than two rings of order $p^2$ (with unity), is the group of units of these rings known?

Two other easy examples are $\mathbb{F}_{p^2}$ and $\mathbb{F}_p[x]/x^2$. — Qiaochu Yuan, Dec 05 '16 at 08:11
In $F_p[x]/(x^2)$, since $x$ is a zero divisor, so are all the multiples of $x$; so units are of the form $a+bx$ with $a$ non-zero in $F_p$, am I right? — p Groups, Dec 05 '16 at 08:19
See also https://oeis.org/A037291, https://oeis.org/A027623, https://oeis.org/A037289 — Watson, Dec 05 '16 at 08:44

score 1 · Accepted Answer · answered Dec 05 '16 at 08:50

In the paper

Benjamin Fine, Classification of Finite Rings of Order $p^2$, Mathematics Magazine, vol. 66, NO. 4, October 1993, p.248-252,

finite rings (possibly without unity) are classified. Up to isomorphism, there are exactly 11 rings of order $p^2$.

To answer your question, one could take these 11 cases and see which ones contain a unity. A quick check (which would benefit from double-checking) seems to indicate that the only rings with unity in this list are the ones already mentioned in the question and comments: $\mathbb{Z}_{p^2}, \mathbb{Z}_p \times \mathbb{Z}_p, \mathbb{F}_{p^2}$ and $\mathbb{F}_p[x]/(x^2)$.

Rings of order $p^2$ and some related information

1 Answers1