Naj bo ▫$R$▫ kolobar z množico vseh nilpotentov ▫$\operatorname{Nil}(R)$▫. Dokažemo, da so naslednje trditve ekvivalentne: (i) ▫$\operatorname{Nil}(R)$▫ je aditivno zaprta, (ii) ...▫$\operatorname{Nil}(R)$▫ je multiplikativno zaprta in ▫$R$▫ zadošča Köthejevi domnevi, (iii) ▫$\operatorname{Nil}(R)$▫ je zaprta za operacijo ▫$x \circ y = x + y - xy$▫, (iv) ▫$\operatorname{Nil}(R)$▫ je podkolobar kolobarja ▫$R$▫. Podamo nekaj aplikacij in primerov kolobarjev s temi lastnostmi, s poudarkom na določenih razredih izmenljivih in čistih kolobarjev.
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