On the existence of nonzero injective covers and projective envelopes of modules

Abstract

In general, the injective cover (projective envelope) of a simple module can be zero. A ring R is called a weakly left V-ring (strongly left Kasch ring) if every simple left R -module has a nonzero injective cover (projective envelope). It is proven that every nonzero left R -module has a nonzero injective cover if and only if R is a left Artinian weakly left V-ring. Dually, every nonzero left R -module has a nonzero projective envelope if and only if R is a left perfect right coherent strongly left Kasch ring. Some related rings and examples are considered.

Traveling Wave Solutions for Nonlinear Differential-Difference Equations of Rational Types

“…We are aware of the fact that not all rational DDEs can be treated with our approach presented here. Thus, as a future task, our plan will be the investigation of exact solutions of such equations via some recent powerful techniques (for example, the modified (G /G)-expansion method, [32] the extended tanh-coth expansion method, [33] the so-called new method, [34] auxiliary ordinary differential equation method, [35] the extended (G /G)-expansion method [35] ), which have been proven to be useful in solving problems of applied mathematical and physical sciences.…”