▁Mor ita ▁equivalent ▁if ▁and ▁only ▁if ▁there ▁are ▁b im od ules ▁R MS ▁and ▁S NR ▁such ▁that ▁ ▁as ▁( R , R ) ▁b im od ules ▁and ▁ ▁as ▁( S , S ) ▁b im od ules . ▁ ▁Moreover , ▁N ▁and ▁M ▁are ▁related ▁via ▁an ▁( S , R ) ▁b im od ule ▁isomorphism : ▁. ▁ ▁More ▁con cret ely , ▁two ▁rings ▁R ▁and ▁S ▁are ▁Mor ita ▁equivalent ▁if ▁and ▁only ▁if ▁ ▁for ▁a ▁pro generator ▁module ▁PR , ▁which ▁is ▁the ▁case ▁if ▁and ▁only ▁if ▁▁▁ ▁( is omorphism ▁of ▁rings ) ▁for ▁some ▁positive ▁integer ▁n ▁and ▁full ▁ide mp ot ent ▁e ▁in ▁the ▁matrix ▁ring ▁M n ( R ). ▁ ▁It ▁is ▁known ▁that ▁if ▁R ▁is ▁Mor ita ▁equivalent ▁to ▁S , ▁then ▁the ▁ring ▁C ( R ) ▁is ▁is omorphic ▁to ▁the ▁ring ▁C ( S ), ▁where ▁C (- ) ▁denotes ▁the ▁center ▁of ▁the ▁ring , ▁and ▁further more ▁R / J ( R ) ▁is ▁Mor ita ▁equivalent ▁to ▁S / J ( S ), ▁where ▁J (- ) ▁denotes ▁the ▁Jacob son ▁radical . ▁ ▁While ▁is omorphic ▁rings ▁are ▁Mor ita ▁equivalent , ▁Mor ita ▁equivalent ▁rings ▁can ▁be ▁non is omorphic . ▁ ▁An ▁easy ▁example ▁is ▁that ▁a ▁division ▁ring ▁D ▁is ▁Mor ita ▁equivalent ▁to ▁all ▁of ▁its ▁matrix ▁rings ▁M n ( D ), ▁but ▁cannot ▁be ▁is omorphic ▁when ▁n