On ADS$^{\circledast}$-(pre)covers and ADS$^{\circledast}$-(pre)envelopes modules

The objective of this paper is to study the question of when modules have ADS$^{\circledast}$-preenvelopes and covers. It is proved that every ADS$^{\circledast}$ module over a ring $R$ is projective if and only if every right $R$-module has an ADS$^{\circledast}$-(pre)envelope (an ADS$^{\circledast}$-(pre)cover). We also introduce a generalization of ADS$^{\circledast}$ modules in terms of their invariance under certain automorphisms of their envelopes.