In the paper of {\it W. C. Waterhouse} [Am. Math. Mon. 71, 449-450 (1964)] it is proved that if a ring has cyclic additive group $C$, then up to an isomorphism $R$ has a presentation $R_d=\langle g$; $mg=0$, $g^2=dg\rangle$ where $d$ is a divisor of $m$.
In this paper the author is looking at the rings with exactly $p^2$ elements, where $p$ is a prime. He proves that there are exactly 11 non-isomorphic rings with $p^2$ elements and provides in theorem 2 a complete list, providing the presentation by generators and relations. The approach is absolutely elementary.
If the reader is interested in learning more about finite rings of higher order we would recommend the book of {\it B. R. McDonald} “Finite rings with identity” (1974;

Zbl 0294.16012); see also Corollary 3 to Theorem 8 in the reviewer’s paper [Commun. Algebra 15, 2327-2348 (1987;

Zbl 0635.16012)].