▁Mad h ya ▁Pr adesh ▁Category : B har ati ya ▁Jan ata ▁Party ▁polit icians ▁from ▁Mad h ya ▁Pr adesh ▁Category : Pe ople ▁from ▁B hind <0x0A> </s> ▁In ▁abstract ▁algebra , ▁Mor ita ▁equivalence ▁is ▁a ▁relationship ▁defined ▁between ▁rings ▁that ▁pres erves ▁many ▁ring - the or etic ▁properties . ▁It ▁is ▁named ▁after ▁Japanese ▁math ematic ian ▁Ki iti ▁Mor ita ▁who ▁defined ▁equivalence ▁and ▁a ▁similar ▁notion ▁of ▁du ality ▁in ▁ 1 9 5 8 . ▁ ▁Mot iv ation ▁ ▁R ings ▁are ▁commonly ▁studied ▁in ▁terms ▁of ▁their ▁modules , ▁as ▁modules ▁can ▁be ▁viewed ▁as ▁representations ▁of ▁rings . ▁Every ▁ring ▁R ▁has ▁a ▁natural ▁R - module ▁structure ▁on ▁itself ▁where ▁the ▁module ▁action ▁is ▁defined ▁as ▁the ▁multiplication ▁in ▁the ▁ring , ▁so ▁the ▁approach ▁via ▁modules ▁is ▁more ▁general ▁and ▁gives ▁useful ▁information . ▁Because ▁of ▁this , ▁one ▁often ▁studies ▁a ▁ring ▁by ▁studying ▁the ▁category ▁of ▁modules ▁over ▁that ▁ring . ▁Mor ita ▁equivalence ▁takes ▁this ▁view point ▁to ▁a ▁natural ▁conclusion ▁by ▁defining ▁rings ▁to ▁be ▁Mor ita ▁equivalent ▁if ▁their ▁module ▁categories ▁are ▁equivalent . ▁This ▁notion ▁is ▁of ▁interest ▁only ▁when ▁dealing ▁with ▁non comm ut ative ▁rings , ▁since ▁it ▁can ▁be ▁shown ▁that ▁two ▁commut ative ▁rings ▁are ▁Mor ita ▁equivalent ▁if ▁and ▁only ▁if ▁they ▁are ▁is omorphic . ▁ ▁Definition ▁ ▁Two ▁rings ▁R ▁and ▁S ▁( associ ative , ▁with ▁ 1 ) ▁are ▁said ▁to ▁be ▁( Mor