Dergiler / Turkish Journal of Mathematics / 2011 / Cilt: 35 - Sayı: 2
Nilpotent elements and reduced rings
- Sayfa
- 341–353
- DOI
- —
Özet
In this paper, we show the following results: (1) R is a min-leftsemicentral ring if and only if eR(1-e)Re=0 for all e \in MEl(R); (2) Quasi-normal rings, NI rings and weakly reversible rings are all min-leftsemicentral ring; (3) R is left MC2 ring if and only if aRe=0 implies eRa=0 for all e \in MEl(R) and a \in R if and only if every projective simple left R-module is MUP-injective; (4) R is reduced if and only if R is n-regular and quasi-normal if and only if R is n-regular and weakly reversible; (5) R is strongly regular if and only if R is regular and quasi-normal if and only if R is regular and weakly reversible.
Abstract
In this paper, we show the following results: (1) R is a min-leftsemicentral ring if and only if eR(1-e)Re=0 for all e \in MEl(R); (2) Quasi-normal rings, NI rings and weakly reversible rings are all min-leftsemicentral ring; (3) R is left MC2 ring if and only if aRe=0 implies eRa=0 for all e \in MEl(R) and a \in R if and only if every projective simple left R-module is MUP-injective; (4) R is reduced if and only if R is n-regular and quasi-normal if and only if R is n-regular and weakly reversible; (5) R is strongly regular if and only if R is regular and quasi-normal if and only if R is regular and weakly reversible.