On Extensions of Weakly Primitive Rings

If R is a ring an R-module M is called compressible when it can be embedded in each of its non-zero submodules; and M is called monoform if each partial endomorphism N → M, N ⊆ M, is either zero or monic. The ring R is called (left) weakly primitive if it has a faithful monoform compressible left module. It is known that a version of the Jacobson density theorem holds for weakly primitive rings [4], and that weak primitivity is a Mori ta invariant and is inherited by a variety of subrings and matrix rings. The purpose of this paper is to show that weak primitivity is preserved under formation of polynomials, rings of quotients, and group rings of torsion-free abelian groups. The key result is that R[x] is weakly primitive when R is (Theorem 1).