Elementary equivalence and isomorphisms of locally nilpotent matrix groups and rings

摘要

Elementary equivalence and isomorphisms of locally nilpotent matrix groups and rings was reported. A fundamental correspondence between the elementary properties of the unitriangular group of degree 3 and the coefficient ring was detected. The relationship between ring derivations and endomorphisms is revealed by various lemma. The various theorems reveal a relationship between the elementary equivalence of the associated rings with the elementary properties of the base rings. Derivations in a ring are trivial Lie and Jordan derivations, and isomorphisms and anti-isomorphisms between rings are trivial Jordan isomorphisms. The elementary equivalence of coefficient rings can be easily transferred to uni triangular groups of the same degree and to the associated Lie and Jordan rings.