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Classification of finite rings of order ${p}^{2}$. (English) Zbl 0838.16011

In the paper of 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 ${R}_{d}=〈g$; $mg=0$, ${g}^{2}=dg〉$ 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 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)].

##### MSC:
 16P10 Finite associative rings and finite-dimensional algebras 13M05 Structure of finite commutative rings 16S15 Finite generation, finite presentability, normal forms 11T30 Structure theory of finite fields