ita ) ▁equivalent ▁if ▁there ▁is ▁an ▁equivalence ▁of ▁the ▁category ▁of ▁( left ) ▁modules ▁over ▁R , ▁R - Mod , ▁and ▁the ▁category ▁of ▁( left ) ▁modules ▁over ▁S , ▁S - Mod . ▁It ▁can ▁be ▁shown ▁that ▁the ▁left ▁module ▁categories ▁R - Mod ▁and ▁S - Mod ▁are ▁equivalent ▁if ▁and ▁only ▁if ▁the ▁right ▁module ▁categories ▁Mod - R ▁and ▁Mod - S ▁are ▁equivalent . ▁Further ▁it ▁can ▁be ▁shown ▁that ▁any ▁fun ctor ▁from ▁R - Mod ▁to ▁S - Mod ▁that ▁yields ▁an ▁equivalence ▁is ▁automatically ▁add itive . ▁ ▁Ex amples ▁ ▁Any ▁two ▁is omorphic ▁rings ▁are ▁Mor ita ▁equivalent . ▁ ▁The ▁ring ▁of ▁n - by - n ▁matrices ▁with ▁elements ▁in ▁R , ▁denoted ▁M n ( R ), ▁is ▁Mor ita - equ ivalent ▁to ▁R ▁for ▁any ▁n ▁> ▁ 0 . ▁Notice ▁that ▁this ▁general izes ▁the ▁classification ▁of ▁simple ▁art in ian ▁rings ▁given ▁by ▁Art in – W ed der burn ▁theory . ▁To ▁see ▁the ▁equivalence , ▁notice ▁that ▁if ▁ ▁X ▁is ▁a ▁left ▁R - module ▁then ▁X n ▁is ▁an ▁M n ( R )- module ▁where ▁the ▁module ▁structure ▁is ▁given ▁by ▁matrix ▁multiplication ▁on ▁the ▁left ▁of ▁column ▁vectors ▁from ▁X . ▁This ▁allows ▁the ▁definition ▁of ▁a ▁fun ctor ▁from ▁the ▁category ▁of ▁left ▁R - modules ▁to ▁the ▁category ▁of ▁left ▁M n ( R )- modules . ▁The ▁inverse ▁fun ctor ▁is ▁defined