Is there a classification of the group of units in every ring of order $p^2$? Let $R$ be a finite ring (with unity) of order $p^2$, where $p$ is a prime.
I know that there are exactly 11 rings of this order.
My question is the following:
Is there a classification of the group of units in every ring of order $p^2$?
If the answer is yes, please recommend a reference.
Thanks.
 A: Only 4 of the 11 rings of order $p^2$ have a unity.
$$
\mathbb{Z}/(p^2)
$$
$$
\mathbb{F}_{p^2}
$$
$$
\mathbb{F}_{p}\times \mathbb{F}_{p}
$$
$$
\mathbb{F}_{p}[X]/(X^2)
$$
See answers to the following question for an explanation:
Classifying Unital Commutative Rings of Order $p^2$
For case 1: the group of units is $C_{p^2-p}$
Source:  https://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n#Cyclic_case
For case 2: the group of units is $C_{p^2-1}$
Source: Finite subgroups of the multiplicative group of a field are cyclic
For case 3: the group of units is $C_{p-1}\times C_{p-1}$
Source: Group of units of direct sum of rings is isomorphic to direct sum of the groups of units
For case 4: an element $a+bX+(X^2)$ in this ring is a unit iff $a\neq 0$, and the group is again cyclic $C_{p^2-p}$.
Sources:
Units in polynomial quotient ring
Is the group of units of a finite ring cyclic?
https://www.jstor.org/stable/2373134?seq=1
A: If $\{0\}$ is maximal then its a field of $p^2$ elements and it's known that the units are the cyclic group of order $p^2-1$.
If $\{0\}$ is not maximal, then there is a maximal ideal of order $p$, call it $M$.  If $M$ is unique, then $R\setminus M$ is the set of units, and as a group of order $p^2-p$.  This case breaks further down into the case where the characteristic is $p$ or the characteristic is $p^2$.
If $M$ is not unique, then there's another maximal ideal $M'$ such that $M\cap M'=\{0\}$ (arguing by orders.). In this case $R\cong F_p\times F_p$, and the group of units is obviously $C_{p-1}\times C_{p-1}$.
