Isomorphisms of certain locally nilpotent finitary groups and associated rings

For any chain Γ the ring NT(Γ, K) of all finitary Γ-matrices ∥a_ij∥_i,jεΓ over an associative ring K with zeros on and above the main diagonal is locally nilpotent and hence radical. If R′ = NT(Γ′, K′), R = NT(Γ, K) and either |Γ| < ∞ or K is a ring with no zero-divisors, then isomorphisms between rings R and R′, their adjoint groups and associated Lie rings are described. chain, finitary matrix, radical ring, adjoint group, associated Lie ring, isomorphism.