▁than ▁cov ari ant . ▁This ▁theory , ▁though ▁similar ▁in ▁form , ▁has ▁significant ▁differences ▁because ▁there ▁is ▁no ▁du ality ▁between ▁the ▁categories ▁of ▁modules ▁for ▁any ▁rings , ▁although ▁dual ities ▁may ▁exist ▁for ▁sub categories . ▁In ▁other ▁words , ▁because ▁infinite ▁dimensional ▁modules ▁are ▁not ▁generally ▁ref lex ive , ▁the ▁theory ▁of ▁dual ities ▁applies ▁more ▁easily ▁to ▁finit ely ▁generated ▁al gebras ▁over ▁no ether ian ▁rings . ▁Perhaps ▁not ▁surprising ly , ▁the ▁criter ion ▁above ▁has ▁an ▁analog ue ▁for ▁dual ities , ▁where ▁the ▁natural ▁isomorphism ▁is ▁given ▁in ▁terms ▁of ▁the ▁hom ▁fun ctor ▁rather ▁than ▁the ▁tensor ▁fun ctor . ▁ ▁Mor ita ▁equivalence ▁can ▁also ▁be ▁defined ▁in ▁more ▁struct ured ▁situations , ▁such ▁as ▁for ▁sym ple ctic ▁gr ou po ids ▁and ▁C * - al gebras . ▁In ▁the ▁case ▁of ▁C * - al gebras , ▁a ▁stronger ▁type ▁equivalence , ▁called ▁strong ▁Mor ita ▁equivalence , ▁is ▁needed ▁to ▁obtain ▁results ▁useful ▁in ▁applications , ▁because ▁of ▁the ▁additional ▁structure ▁of ▁C * - al gebras ▁( coming ▁from ▁the ▁invol utive ▁* - operation ) ▁and ▁also ▁because ▁C * - al gebras ▁do ▁not ▁necessarily ▁have ▁an ▁identity ▁element . ▁ ▁Sign ific ance ▁in ▁K - theory ▁ ▁If ▁two ▁rings ▁are ▁Mor ita ▁equivalent , ▁there ▁is ▁an ▁induced ▁equivalence ▁of ▁the ▁respective ▁categories ▁of ▁project ive ▁modules ▁since ▁the ▁Mor ita ▁equival ences ▁will ▁preserve ▁exact ▁sequences ▁( and ▁hence