▁by ▁real izing ▁that ▁for ▁any ▁M n ( R )- module ▁there ▁is ▁a ▁left ▁R - module ▁X ▁such ▁that ▁the ▁M n ( R )- module ▁is ▁obtained ▁from ▁X ▁as ▁described ▁above . ▁ ▁C riteria ▁for ▁equivalence ▁ ▁Equ ival ences ▁can ▁be ▁character ized ▁as ▁follows : ▁if ▁F : R - Mod ▁ ▁S - Mod ▁and ▁G : S - Mod ▁R - Mod ▁are ▁add itive ▁( cov ari ant ) ▁fun ctors , ▁then ▁F ▁and ▁G ▁are ▁an ▁equivalence ▁if ▁and ▁only ▁if ▁there ▁is ▁a ▁bal anced ▁( S , R )- b im od ule ▁P ▁such ▁that ▁ ▁SP ▁ ▁and ▁ ▁PR ▁ ▁are ▁finit ely ▁generated ▁project ive ▁gener ators ▁and ▁there ▁are ▁natural ▁is omorph isms ▁of ▁the ▁fun ctors ▁, ▁and ▁of ▁the ▁fun ctors ▁ ▁F initely ▁generated ▁project ive ▁gener ators ▁are ▁also ▁sometimes ▁called ▁pro gener ators ▁for ▁their ▁module ▁category . ▁ ▁For ▁every ▁right - ex act ▁fun ctor ▁F ▁from ▁the ▁category ▁of ▁left - R ▁modules ▁to ▁the ▁category ▁of ▁left - S ▁modules ▁that ▁comm utes ▁with ▁direct ▁sums , ▁a ▁theorem ▁of ▁hom ological ▁algebra ▁shows ▁that ▁there ▁is ▁a ▁( S , R )- b im od ule ▁E ▁such ▁that ▁the ▁fun ctor ▁ ▁is ▁naturally ▁is omorphic ▁to ▁the ▁fun ctor ▁. ▁Since ▁equival ences ▁are ▁by ▁necessity ▁exact ▁and ▁comm ute ▁with ▁direct ▁sums , ▁this ▁implies ▁that ▁R ▁and ▁S ▁are