Unit-Regularity of Regular Nilpotent Elements

2015/09/26 by Khurana, Dinesh
#16E50 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1509.07944

Abstract

Let a be a regular element of a ring R. If either K:=\rmr_R(a) has the exchange property or every power of a is regular, then we prove that for every positive integer n there exist decompositions R_R = K ⊕ X_n ⊕ Y_n = E_n ⊕ X_n ⊕ aY_n, where Y_n ⊆ aⁿR and E_n ≅ 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.

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