Projective maximal submodules of extending regular modules

Document Type: Research Paper

Author

Yasouj University

Abstract

We show  that a projective maximal submodule of a
finitely generated, regular, extending module is a direct
summand. Hence, every finitely generated, regular, extending
module with projective maximal submodules is semisimple. As a
consequence, we observe that every regular, hereditary, extending
module is semisimple. This generalizes and simplifies a result of
  Dung and   Smith. As another consequence, we observe that
every right continuous ring, whose maximal right ideals are
projective, is semisimple Artinian. This generalizes some results
of   Osofsky and   Karamzadeh. We also observe that
four classes of rings, namely right $\aleph_0$-continuous rings,
right continuous rings, right $\aleph_0$-continuous regular rings
and right continuous regular rings are not axiomatizable.

Keywords