Density Theorems for Super-Rings
by
Sergei Limarenko

The main purpose of this work is to consider some questions concerning density problems for associative super-rings. The work contains such classical results as Jacobson and Amitsur theorems and also two extended density theorems analogous to those for rings. One of the main results is

Theorem Let R=R₀+R₁ be a weak primitive super-ring, M=M₀+M₁ a critically compressible right R-supermodule. Let [`M] be a quasi-injective hull of M and \Delta = End([`(M_R)]) ([`M]=\DeltaM). Then we have the following conditions.
1. Let v_1j, ... , v_kj in [`M] be linearly independent over \Delta<sub>0</sub>. Then [`M]=\DeltaM and (1) there exists a_\alpha in \Delta_\alpha such that for any n_1\eta, ... , n_k\eta in M_\eta there exists r_\rho in R_\rho (\rho = \alpha+\eta+j) such that a_\alpha n_i\eta=v_ijr_\rho, i=1, ... , k.
2. Given any \tau_\tau in End_\Delta([`M]) and any elements m_1\mu, ... , m_t\mu in M linearly independent over \Delta₀ there exist r_\rho, s_\tau+\rho in R with m_i\mu\tau_\tau r_\rho = m_i\mus_\tau+\rho and 0 =/= m_i\mur_\rho in \Deltam_i\mu for alli.

Date received: November 14, 2000