Communications of the
Korean Mathematical Society CKMS

This article concerns the normal property of elements on Jacobson and nil radicals which are generalizations of commutativity. A ring is said to be {\it right njr} if it satisfies the normal property on the Jacobson radical. Similarly a ring is said to be {\it right nunr} (resp., {\it right nlnr}) if it satisfies the normal property on the upper (resp., lower) nilradical. We investigate the relations between right duo property and the normality on Jacobson (nil) radicals. Related examples are investigated in the procedure of studying the structures of right njr, nunr, and nlnr rings.

Keywords: right njr, right nunr, right nlnr, Jacobson radical, upper nilradical, lower nilradical, Abelian ring, right duo ring, reduced ring, matrix ring, polynomial ring

MSC numbers: 16N20, 16N40, 16U80