Unit-Regularity of Regular Nilpotent Elements

Abstract

Let a be a regular element of a ring R. If either K:=r _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

$$R_{R} = K \oplus X_{n} \oplus Y_{n} = E_{n} \oplus X_{n} \oplus aY_{n}, $$

where $Y_{n} \subseteq a^{n}R$ and E _n ≅R/a R. 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.