Geodesic
Enumeration of $\mathcal D$-invariant ideals of the ring $\mathrm{R_n(K,J)}$
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 4 (2011) no. 3, pp. 332-343.

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Let $K$ be a local ring of the main ideal with a nilpotent maximal ideal $J$. The paper is devoted to finished of solution of problem enumeration of ideals of the ring $K$ of $n\times n$ matrices with coefficients of $J$ on the main diagonal and above it.
Keywords: combinatorial identities, method of coefficients, enumeration of lattice, ring theory. 
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Maxim N. Davletshin. Enumeration of $\mathcal D$-invariant ideals of the ring $\mathrm{R_n(K,J)}$. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 4 (2011) no. 3, pp. 332-343. http://geodesic.mathdoc.fr/item/JSFU_2011_4_3_a6/

[1] V. M. Levchuk, “Podgruppy unitreugolnykh grupp”, Izvestiya AN SSSR, Ser. matem., 38:6 (1974), 1202–1220 | MR | Zbl

[2] G. P. Egorychev, Integralnye predstavleniya i vychislenie kombinatornykh summ, Nauka, Novosibirsk, 1977 ; English: Transl. of Math. Monographs, 59, AMS, 1984 ; 2-nd Ed., 1989 | Zbl | MR | Zbl

[3] B. A. Tolasov, “The number of normal subgroups of triangular group that are contained in the unitriangular subgroup”, Algebra and Number Theory, v. 2, Kabardino-Balkarsk. Gos. Univ., Nalchik, 1977, 122–126 | MR

[4] V. M. Levchuk, “Svyazi unitreugolnoi gruppy s nekotorymi koltsami”, Algebra i logika, 5 (1976), 348–360

[5] R. Dubish, S. Perlis, “On total nilpotent algebras”, Amer. J. Math., 73:3 (1951), 439–452 | DOI | MR

[6] G. P. Egorychev, V. M. Levchuk, “Enumeration of characteristic subgroups of unipotent Lie-type groups”, Algebra, eds. Yu. L. Ershov, E. I. Khukhro, V. M. Levchuk, N. D. Podufalov, Walter de Gruyter, Berlin–New York, 1996, 49–62 | MR | Zbl

[7] G. P. Egorychev, V. M. Levchuk, “Perechislitelnye problemy dlya grupp i algebr tipa Li”, Dokl. RAN, 330:3 (1993), 291–293 | MR | Zbl

[8] G. P. Egorychev, V. M. Levchuk, “Enumeration in the Chevalley algebras”, ACM SIGSAM Bulletin, 35 (2001), 20–34 | DOI | Zbl

[9] F. Kuzucuoglu, V. M. Levchuk, “Ideals of some matrix rings”, Commun. in Algebra, 28:7 (2000), 3503–3513 | DOI | MR | Zbl

[10] V. M. Levchuk, G. C. Suleimanova, “Normalnoe stroenie prisoedinennoi gruppy v radikalnykh koltsakh $R_n(K,J)$”, Sib. matem. zhurn., 43:2 (2002), 419–437 | MR | Zbl

[11] M. N. Davletshin, G. P. Egorychev, V. M. Levchuk, “Kombinatornaya formula dlya $D$-invariantnykh idealov koltsa $R_n(K,J)$”, Materialy Vserossiiskoi konferentsii Algebra, logika i metodika obucheniya matematike, Krasnoyar. gos. ped. un-t im. V. P. Astafeva, Krasnoyarsk, 2010, 23–32

[12] Kourovskaya tetrad. Nereshennye voprosy teorii grupp, 14-e izd., IM SO RAN, Novosibirsk, 1999 | MR

[13] V. M. Petrogradsky, “Growth of finitely generated polynilpotent Lie algebras and groups, generalized partitions, and functions analytic in the unit circle”, Intern. J. Algebra Comput., 9 (1999), 179–212 | DOI | MR | Zbl