i -) her edit ary ▁right ▁( or ▁left ) ▁n ons ing ular ▁right ▁( or ▁left ) ▁coh er ent ▁semi primary , ▁right ▁( or ▁left ) ▁perfect , ▁semi per fect ▁ ▁sem il ocal ▁ ▁Ex amples ▁of ▁properties ▁which ▁are ▁not ▁Mor ita ▁invariant ▁include ▁commut ative , ▁local , ▁reduced , ▁domain , ▁right ▁( or ▁left ) ▁Gold ie , ▁Fro ben ius , ▁invariant ▁basis ▁number , ▁and ▁D ed ek ind ▁finite . ▁ ▁There ▁are ▁at ▁least ▁two ▁other ▁tests ▁for ▁determ ining ▁whether ▁or ▁not ▁a ▁ring ▁property ▁ ▁is ▁Mor ita ▁invariant . ▁ ▁An ▁element ▁e ▁in ▁a ▁ring ▁R ▁is ▁a ▁full ▁ide mp ot ent ▁when ▁e 2   =   e ▁and ▁Re R   =   R . ▁▁ ▁is ▁Mor ita ▁invariant ▁if ▁and ▁only ▁if ▁whenever ▁a ▁ring ▁R ▁satisfies ▁, ▁then ▁so ▁does ▁e Re ▁for ▁every ▁full ▁ide mp ot ent ▁e ▁and ▁so ▁does ▁every ▁matrix ▁ring ▁M n ( R ) ▁for ▁every ▁positive ▁integer ▁n ; ▁or ▁ ▁is ▁Mor ita ▁invariant ▁if ▁and ▁only ▁if : ▁for ▁any ▁ring ▁R ▁and ▁full ▁ide mp ot ent ▁e ▁in ▁R , ▁R ▁satisfies ▁ ▁if ▁and ▁only ▁if ▁the ▁ring ▁e Re ▁satisfies ▁. ▁ ▁Further ▁directions ▁ ▁D ual ▁to ▁the ▁theory ▁of ▁equival ences ▁is ▁the ▁theory ▁of ▁dual ities ▁between ▁the ▁module ▁categories , ▁where ▁the ▁fun ctors ▁used ▁are ▁cont rav ari ant ▁rather