PROJECTIVE MAXIMAL SUBMODULES OF EXTENDING REGULAR MODULES

We show that a projective maximal submodule of a nitely 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 N0-continuous rings, right continuous rings, right N0-continuous regular rings and right continuous regular rings are not axiomatizable.