Результаты исследований: Научные публикации в периодических изданиях › статья › Рецензирование
Divisible Rigid Groups. Algebraic Closedness and Elementary Theory. / Romanovskii, N. S.
в: Algebra and Logic, Том 56, № 5, 01.11.2017, стр. 395-408.Результаты исследований: Научные публикации в периодических изданиях › статья › Рецензирование
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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 -
ID: 9672149