(Simple) Virtually Regular Modules

As a proper generalization of regular modules,the concepts of virtually regular modules and rings are introduced in Chapter 3.A module M is said to be a virtually regular module if every finitely generated submodule of M is isomorphic to a direct summand of M.If_RR is a virtually regular module,then we say that R is a left virtually regular ring.Basic properties of these modules and rings are studied.It is shown that a ring R is a left(strongly)virtually regular ring if and only if every(projective)free left R-module is a(semicompletely)virtually regular module.The concept of simple virtually regular modules is posed in Chapter 4,basic properties of simple virtually regular modules are investigated.Under some conditions,a direct summand of a simple virtually regular module is a simple virtually regular module.V-rings are charac-terized by using simple virtually regular modules.It is shown that a ring R is a left V-ring if and only if every left R-module is a simple virtually regular module if and only if every left finitely cogenerated R-module is a simple virtually regular module.