▁ ▁External ▁links ▁▁ ▁Category : 1 9 9 0 ▁birth s ▁Category : L iving ▁people ▁Category : Russ ian ▁male ▁bad m inton ▁players <0x0A> </s> ▁In ▁category ▁theory , ▁an ▁abstract ▁mathematical ▁discipline , ▁a ▁nod al ▁decomposition ▁of ▁a ▁morph ism ▁ ▁is ▁a ▁representation ▁of ▁ ▁as ▁a ▁product ▁, ▁where ▁ ▁is ▁a ▁strong ▁ep im orphism , ▁ ▁a ▁b im orphism , ▁and ▁ ▁a ▁strong ▁mon omorphism . ▁ ▁Un iqu eness ▁and ▁not ations ▁▁▁ ▁If ▁exists , ▁the ▁nod al ▁decomposition ▁is ▁unique ▁up ▁to ▁an ▁isomorphism ▁in ▁the ▁following ▁sense : ▁for ▁any ▁two ▁ ▁nod al ▁decom pos itions ▁ ▁and ▁ ▁there ▁exist ▁is omorph isms ▁ ▁and ▁ ▁such ▁that ▁▁▁▁▁▁▁▁▁▁ ▁This ▁property ▁just ifies ▁some ▁special ▁not ations ▁for ▁the ▁elements ▁of ▁the ▁nod al ▁decomposition : ▁▁▁ ▁– ▁here ▁ ▁and ▁ ▁are ▁called ▁the ▁nod al ▁co image ▁of ▁, ▁ ▁and ▁ ▁the ▁nod al ▁image ▁of ▁, ▁and ▁ ▁the ▁nod al ▁reduced ▁part ▁of ▁. ▁ ▁In ▁these ▁not ations ▁the ▁nod al ▁decomposition ▁takes ▁the ▁form ▁ ▁Connection ▁with ▁the ▁basic ▁decomposition ▁in ▁pre - abel ian ▁categories ▁▁ ▁In ▁a ▁pre - abel ian ▁category ▁ ▁each ▁morph ism ▁ ▁has ▁a ▁standard ▁decomposition ▁▁ ▁, ▁called ▁the ▁basic ▁decomposition ▁( here ▁, ▁, ▁and ▁ ▁are ▁respectively ▁the ▁image , ▁the ▁co image ▁and ▁the ▁reduced ▁part ▁of ▁the ▁morph ism ▁). ▁▁ ▁If ▁a ▁morph ism ▁ ▁in