Abstract:
We describe the structure of the finite primary rings of principal ideals; we prove that every such ring is the factor-ring of the ring of integers of a finite extension of the field of rational p-adic numbers; we touch on the problem of the number of nonisomorphic rings of this type with a fixed number of elements.
Citation:
A. A. Nechaev, “On the structure of finite commutative rings with an identity”, Mat. Zametki, 10:6 (1971), 679–688; Math. Notes, 10:6 (1971), 840–845
\Bibitem{Nec71}
\by A.~A.~Nechaev
\paper On the structure of finite commutative rings with an identity
\jour Mat. Zametki
\yr 1971
\vol 10
\issue 6
\pages 679--688
\mathnet{http://mi.mathnet.ru/mzm9747}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=291153}
\zmath{https://zbmath.org/?q=an:0237.12014}
\transl
\jour Math. Notes
\yr 1971
\vol 10
\issue 6
\pages 840--845
\crossref{https://doi.org/10.1007/BF01146443}
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This publication is cited in the following 8 articles:
O. A. Kozlitin, “Periodicheskie svoistva mnogomernykh polinomialnykh preobrazovanii nad koltsom Galua – Eizenshteina”, Matem. vopr. kriptogr., 13:1 (2022), 69–99
O. A. Kozlitin, “Estimate of the maximal cycle length in the graph of polynomial transformation of Galois–Eisenstein ring”, Discrete Math. Appl., 28:6 (2018), 345–358
A. A. Nechaev, V. N. Tsypyschev, “Artinian bimodule with quasi-Frobenius bimodule of translations”, Discrete Math. Appl., 29:2 (2019), 103–119
A. V. Abornev, “Recursively-generated permutations of a binary space”, Matem. vopr. kriptogr., 5:2 (2014), 7–20
V. L. Kurakin, “Similarity invariants for matrices over a commutative Artinian chain ring”, Math. Notes, 80:3 (2006), 387–395
D. A. Mikhailov, A. A. Nechaev, “Solving systems of polynomial equations over Galois–Eisenstein rings with the use of the canonical generating systems of polynomial ideals”, Discrete Math. Appl., 14:1 (2004), 41–73
T. G. Gazaryan, “An example of non-isomorphic commutative chain rings”, Russian Math. Surveys, 47:3 (1992), 174–175
A. A. Nechaev, “Finite principal ideal rings”, Math. USSR-Sb., 20:3 (1973), 364–382
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