  >   1 . ▁ ▁In ▁the ▁special ▁case ▁of ▁commut ative ▁rings , ▁Mor ita ▁equivalent ▁rings ▁are ▁actually ▁is omorphic . ▁ ▁This ▁follows ▁immediately ▁from ▁the ▁comment ▁above , ▁for ▁if ▁R ▁is ▁Mor ita ▁equivalent ▁to ▁S , ▁. ▁ ▁Properties ▁preserved ▁by ▁equivalence ▁ ▁Many ▁properties ▁are ▁preserved ▁by ▁the ▁equivalence ▁fun ctor ▁for ▁the ▁objects ▁in ▁the ▁module ▁category . ▁ ▁Gener ally ▁speaking , ▁any ▁property ▁of ▁modules ▁defined ▁purely ▁in ▁terms ▁of ▁modules ▁and ▁their ▁hom omorph isms ▁( and ▁not ▁to ▁their ▁underlying ▁elements ▁or ▁ring ) ▁is ▁a ▁categor ical ▁property ▁which ▁will ▁be ▁preserved ▁by ▁the ▁equivalence ▁fun ctor . ▁ ▁For ▁example , ▁if ▁F (- ) ▁is ▁the ▁equivalence ▁fun ctor ▁from ▁R - Mod ▁to ▁S - Mod , ▁then ▁the ▁R ▁module ▁M ▁has ▁any ▁of ▁the ▁following ▁properties ▁if ▁and ▁only ▁if ▁the ▁S ▁module ▁F ( M ) ▁does : ▁inject ive , ▁project ive , ▁flat , ▁faithful , ▁simple , ▁sem is im ple , ▁finit ely ▁generated , ▁finit ely ▁presented , ▁Art in ian , ▁and ▁No ether ian . ▁ ▁Ex amples ▁of ▁properties ▁not ▁necessarily ▁preserved ▁include ▁being ▁free , ▁and ▁being ▁cy clic . ▁ ▁Many ▁ring ▁the or etic ▁properties ▁are ▁stated ▁in ▁terms ▁of ▁their ▁modules , ▁and ▁so ▁these ▁properties ▁are ▁preserved ▁between ▁Mor ita ▁equivalent ▁rings . ▁ ▁Properties ▁shared ▁between ▁equivalent ▁rings ▁are ▁called ▁Mor ita ▁invariant ▁properties . ▁ ▁For ▁example ,