We study whether the projective and injective properties of left $R$-modules can be implied to the special kind of left $R[x]$-modules, especially to the case of inverse polynomial modules and Laurent polynomial modules.
In this paper we compute injective, projective and flat dimensions of inverse polynomial modules as $R[x]$-modules. We also generalize Hom and Ext functors of inverse polynomial modules to any submonoid but we show Tor functor of inverse polynomial modules can be generalized only for a symmetric submonoid.