Divisible Rigid Groups. Algebraic Closedness and Elementary Theory
Divisible Rigid Groups. Algebraic Closedness and Elementary Theory
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可分刚性群。
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发表时间：
2017
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通讯作者：
N. Romanovskii
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作者：
N. Romanovskii
如果群G包含正规级数G = G1 > G2 >…> Gm > Gm+1 = 1，其商Gi/Gi+1是阿贝尔的，并且作为右模（G [G/Gi]）是无扭的，则称群G是刚性的。如果商Gi/Gi+1的元素能被环G [G/Gi]的非零元素整除，则刚性群G可整除。每一个刚性群都嵌入在一个可分群中。我们证明了两个定理。定理1指出群G的三个条件是等价的：G在所有m刚性群的Σm类中是代数闭的；G在类Σm中是存在闭的；G是一个可整除的m刚性群。定理2表明一类可整除的m-刚性群的初等理论是完备的。
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.