Elementary equivalence of rings with finitely generated additive groups

Myasnikov, Alexei; Oger, Francis; Sohrabi, Mahmood

doi:10.1016/j.apal.2018.01.005

Cited by 4 publications

(2 citation statements)

References 14 publications

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“…This ring constitutes an important feature of f , and in some sense it provides an "approximation" to interpreting (in (A, B; f )) multiplication of constant elements from N and M by integer variables. It has been used successfully to study different first order theoretic aspects of different types of structures, including rings whose additive group is finitely generated [32], free algebras [22,23,24], and nilpotent groups [33,34]. For us the most important property of R(f ) is that it is e-interpretable in (A, B; f ): Theorem 3.1 (Theorem 3.5 of [16]).…”

Section: Largest Ring Of Scalars Of Bilinear Maps and Rings Of Algebr...mentioning

confidence: 99%

Diophantine problems in solvable groups

2020

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We study systems of equations in different classes of solvable groups. For each group G in one of these classes we prove that there exists a ring of algebraic integers O that is interpretable in G by systems of equations (e-interpretable). This leads to the conjecture that Z is e-interpretable in G and that the Diophantine problem in G is undecidable. We further prove that Z is e-interpretable in any generalized Heisenberg group and in any finitely generated nonabelian free (solvable-by-nilpotent) group. The latter applies in particular to the case of free solvable groups and to the already known case of free nilpotent groups.

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Section: Largest Ring Of Scalars Of Bilinear Maps and Rings Of Algebr...mentioning

confidence: 99%

Diophantine problems in solvable groups

2020

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“…From bilinear maps to commutative algebra Next, we briefly explain the approach taken in order to pass from possibly non-associative, non-commutative, and nonunitary algebras to algebras of scalars (and similarly for rings). The ideas we present here were introduced by the second author in [26], and they have been used successfully to study different first order theoretic aspects of different types of structures, including rings whose additive group is finitely generated [27], free algebras [16,18,19], and nilpotent groups [28,29].…”

Section: R Is a Simply Graded Lie Algebra Andmentioning

confidence: 99%

Diophantine problems in rings and algebras: undecidability and reductions to rings of algebraic integers

Garreta¹,

Miasnikov²,

Ovchinnikov³

2018

Preprint

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We study systems of equations in different families of rings and algebras. In each such structure R we interpret by systems of equations (e-interpret) a ring of integers O of a global field. The long standing conjecture that Z is always e-interpretable in O then carries over to R, and if true it implies that the Diophantine problem in R is undecidable. The conjecture is known to be true if O has positive characteristic, i.e. if O is not a ring of algebraic integers. As a corollary we describe families of structures where the Diophantine problem is undecidable, and in other cases we conjecture that it is so. In passing we obtain that the first order theory with constants of all the aforementioned structures R is undecidable.

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First-order rigidity of rings satisfying polynomial identities

Greenfeld

2022

Annals of Pure and Applied Logic

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Elementary equivalence of rings with finitely generated additive groups

Cited by 4 publications

References 14 publications

Diophantine problems in solvable groups

Diophantine problems in solvable groups

Diophantine problems in rings and algebras: undecidability and reductions to rings of algebraic integers

First-order rigidity of rings satisfying polynomial identities

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