Unit-Regularity of Regular Nilpotent Elements

الأصل · EN

Let a be a regular element of a ring R. If either K:=rᵣ(a) has the exchange property or every power of a is regular, then we prove that for every positive integer n there exist decompositions Rᵣ = K ⊕ Xₙ ⊕ Yₙ = Eₙ ⊕ Xₙ ⊕ aYₙ, where Yₙ aⁿR and Eₙ R/aR. As applications we get easier proofs of the results that a strongly π-regular ring has stable range one and also that a strongly π-regular element whose every power is regular is unit-regular.