On some rings whose modules have maximal submodules

Abstract:

It is shown that a principal right ideal domain, having the property that every right $R$ module has a maximal submodule must be simple. Strong conditions satisfied by these rings are deduced giving evidence for the conjecture that they must be $V$-rings. We also generalize an example of Faith by showing that a subring of an infinite dimensional full linear ring, which contains the socle of that ring is never a left $V$-ring.