Projective maximal submodules of extending regular modules

Abstract

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