TY - JOUR

T1 - Divisible Rigid Groups. Algebraic Closedness and Elementary Theory

AU - Romanovskii, N. S.

PY - 2017/11/1

Y1 - 2017/11/1

N2 - A group G is said to be rigid if it contains a normal series G = G1 > G2 > … > Gm > Gm+1 = 1, whose quotients Gi/Gi+1 are Abelian and, treated as right ℤ[G/Gi]-modules, are torsion-free. A rigid group G is divisible if elements of the quotient Gi/Gi+1 are divisible by nonzero elements of the ring ℤ[G/Gi]. Every rigid group is embedded in a divisible one. We prove two theorems. Theorem 1 says that the following three conditions for a group G are equivalent: G is algebraically closed in the class Σm of all m-rigid groups; G is existentially closed in the class Σm; G is a divisible m-rigid group. Theorem 2 states that the elementary theory of a class of divisible m-rigid groups is complete.

AB - A group G is said to be rigid if it contains a normal series G = G1 > G2 > … > Gm > Gm+1 = 1, whose quotients Gi/Gi+1 are Abelian and, treated as right ℤ[G/Gi]-modules, are torsion-free. A rigid group G is divisible if elements of the quotient Gi/Gi+1 are divisible by nonzero elements of the ring ℤ[G/Gi]. Every rigid group is embedded in a divisible one. We prove two theorems. Theorem 1 says that the following three conditions for a group G are equivalent: G is algebraically closed in the class Σm of all m-rigid groups; G is existentially closed in the class Σm; G is a divisible m-rigid group. Theorem 2 states that the elementary theory of a class of divisible m-rigid groups is complete.

KW - algebraic closedness

KW - divisible rigid group

KW - elementary theory

KW - SOLUBLE GROUPS

KW - GROUP-RINGS

UR - http://www.scopus.com/inward/record.url?scp=85035793341&partnerID=8YFLogxK

U2 - 10.1007/s10469-017-9461-6

DO - 10.1007/s10469-017-9461-6

M3 - Article

AN - SCOPUS:85035793341

VL - 56

SP - 395

EP - 408

JO - Algebra and Logic

JF - Algebra and Logic

SN - 0002-5232

IS - 5

ER -