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Quasiunitriangular groups

Published online by Cambridge University Press:  12 March 2014

O. V. Belegradek
O. V. Belegradek*
Affiliation:
Kemerovo University, Kemerovo 650043, Russia, E-mail: beleg@kemucnit.kemerovo.su
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For a ring with unit R, which need not be associative, denote the group of upper unitriangular 3 × 3 matrices over R by UT3(R). Let e1 = (1,0,0), e2 = (0,1,0), where (α, β, γ) denotes the matrix

Denote the expanded group (UT3(R), e1, e2) by (R). A. 1. Mal′cev [M] gave an algebraic characterization of the expanded groups of the form (R) as follows. Let h1, h2 be elements of a group H; then (H, h1, h2) is isomorphic to (R), for some R, if and only if

(i) H is 2-step nilpotent;

(ii) CH(hi) are abelian, i = 1,2;

(iii) CH(h1) ∩ CH(h2) = Z(H);

(iv) [CH(h1),h2] = [h1, CH(h2)] = Z(H);

(v) Z(H) is a direct summand in both CH(hi).

(In [M] condition (v) is a bit stronger; the version above is presented in [B2].)

A pair (h1, h2) of elements of a group H is said to be a base if (H, h1, h2) satisfies the conditions (i)–(iv). A. I. Mal′cev [M] found a uniform way of first order interpreting a ring Ring(H, h1, h2) in any group with a base (H, h1, h2); in particular, Ring((R)) ≃ R.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 58 , Issue 1 , March 1993 , pp. 205 - 218
Copyright
Copyright © Association for Symbolic Logic 1993

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References

REFERENCES

[B1]Belegradek, O. V., On the Mal'cev correspondence between rings and groups, Proceedings of the 7th Easter Conference on Model Theory, Wendisch-Rietz(DDR), 1989, Humboldt-Universität zu Berlin, Sektion Mathematik, Seminar berichte Nr. 104, pp. 43–57.Google Scholar
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