On the existence of nonzero injective covers and projective envelopes of modules. (Q2873169)

The existence of different types of envelopes and covers has been studied since 1953 when \textit{B. Eckmann} and \textit{A. Schopf} [Arch. Math. 4, 75-78 (1953; Zbl 0050.25904)] proved that every module has an injective envelope. Whereas injective envelopes and projective covers (if they exist) of a nonzero module are nonzero, it is known that the injective cover (if it exists) of a nonzero module can be zero. \textit{R. Belshoff} and \textit{J. Xu} [Missouri J. Math. Sci. 13, No. 3 (2001; Zbl 1029.16003)] proved that the ring \(R\) is left Artinian if every nonzero left \(R\)-module has a nonzero injective cover, but show that the converse is false.NEWLINENEWLINE In this paper, it is shown that a necessary and sufficient condition for every nonzero left \(R\)-module to have a nonzero injective cover (resp. projective envelope) is that \(R\) is left Artinian and every simple left \(R\)-module has a nonzero injective cover (resp. \(R\) is left perfect right coherent and every simple left \(R\)-module has a nonzero projective envelope). A ring \(R\) is defined to be a weakly left \(V\)-ring (resp. left strongly left Kasch ring) if every simple left \(R\)-module has a nonzero injective cover (resp. projective envelope). Some properties of these and related rings are investigated and a number of examples are given.