Geodesic
Generic types and generic elements in divisible rigid groups
Algebra i logika, Tome 62 (2023) no. 1, pp. 102-113.

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A group $G$ is said to be $m$-rigid if it contains a normal series of the form $$G=G_1>G_2>\ldots>G_m>G_{m+1}=1,$$ whose quotients $G_i/G_{i+1}$ are Abelian and, treated as (right) ${\mathbb{Z}}[G/G_i]$-modules, are torsion-free. A rigid group $G$ is said to be divisible if elements of the quotient $\rho_i(G)/\rho_{i+1}(G)$ are divisible by nonzero elements of the ring ${\mathbb{Z}}[G/\rho_i(G)]$. Previously, it was proved that the theory of divisible $m$-rigid groups is complete and $\omega$-stable. In the present paper, we give an algebraic description of elements and types that are generic over a divisible $m$-rigid group $G$.
Mots-clés : divisible $m$-rigid group
Keywords: generic type, generic element. 
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A. G. Myasnikov; N. S. Romanovskii. Generic types and generic elements in divisible rigid groups. Algebra i logika, Tome 62 (2023) no. 1, pp. 102-113. http://geodesic.mathdoc.fr/item/AL_2023_62_1_a7/

[1] N. S. Romanovskii, “Delimye zhestkie gruppy”, Algebra i logika, 47:6 (2008), 762–776 | MR | Zbl

[2] N. S. Romanovskii, “Neterovost po uravneniyam zhestkikh razreshimykh grupp”, Algebra i logika, 48:2 (2009), 258–279 | MR | Zbl

[3] N. S. Romanovskii, “Neprivodimye algebraicheskie mnozhestva nad delimymi raspavshimisya zhestkimi gruppami”, Algebra i logika, 48:6 (2009), 793–818 | MR | Zbl

[4] A. Myasnikov, N. Romanovskiy, “Krull dimension of solvable groups”, J. Algebra, 324:10 (2010), 2814–2831 | DOI | MR | Zbl

[5] N. S. Romanovskii, “Koproizvedeniya zhestkikh grupp”, Algebra i logika, 49:6 (2010), 803–818

[6] N. S. Romanovskii, “Teorema Gilberta o nulyakh (Nullstellensatz) v algebraicheskoi geometrii nad zhestkimi razreshimymi gruppami”, Izv. RAN. Ser. matem., 79:5 (2015), 201–214 | DOI | MR | Zbl

[7] A. G. Myasnikov, N. S. Romanovskii, “Ob universalnykh teoriyakh zhestkikh razreshimykh grupp”, Algebra i logika, 50:6 (2011), 802–821

[8] N. S. Romanovskii, “Delimye zhestkie gruppy. Algebraicheskaya zamknutost i elementarnaya teoriya”, Algebra i logika, 56:5 (2017), 593–612 | MR | Zbl

[9] A. G. Myasnikov, N. S. Romanovskii, “Delimye zhestkie gruppy. II. Stabilnost, nasyschennost i elementarnye podmodeli”, Algebra i logika, 57:1 (2018), 43–56 | MR | Zbl

[10] N. S. Romanovskii, “Delimye zhestkie gruppy. III. Odnorodnost i eliminatsiya kvantorov”, Algebra i logika, 57:6 (2018), 733–748 | MR

[11] N. S. Romanovskii, “Delimye zhestkie gruppy. IV. Opredelimye podgruppy”, Algebra i logika, 59:3 (2020), 344–366 | MR | Zbl

[12] Z. Sela, “Diophantine geometry over groups. VIII: Stability”, Ann. of Math. (2), 177:3 (2013), 787–868 | DOI | MR | Zbl

[13] D. Marker, Model theory: an introduction, Grad. Texts Math., 217, Springer-Verlag, New York etc., 2002 | MR | Zbl

[14] C. Perin, R. Sklinos, “Forking and JSJ decompositions in the free group”, J. Eur. Math. Soc. (JEMS), 18:9 (2016), 1983–2017 | DOI | MR | Zbl

[15] C. Perin, R. Sklinos, “Forking and JSJ decompositions in the free group. II”, J. Lond. Math. Soc., II. Ser., 102:2 (2020), 796–817 | DOI | MR | Zbl

[16] I. Kherstein, Nekommutativnye koltsa, Mir, M., 1972

[17] I. Lewin, “A note on zero divisors in group-rings”, Proc. Am. Math. Soc., 31:2 (1972), 357–359 | DOI | MR | Zbl

[18] P. H. Kropholler, P. A. Linnell, J. A. Moody, “Applications of a new K-theoretic theorem on soluble group rings”, Proc. Am. Math. Soc., 104:3 (1988), 675–684 | MR | Zbl