some characterizations of the maximal zg-regular ideal in a ring

Let r be an associative ring with identity. a ring r is called zg-regular( resp. strongly zg-regular) if, for every a in r, there exist positive integer n and g in g, such that ang ∈a^ngra^ng (resp. a^ng ∈a^(n+1)gr. in this paper, we shall show that the join of all zg-regular ideals in an arbitrary ring r is a zg-regular ideal, and so there exists a unique maximal zg-regular ideal m = m(r) in r, whose structure we investigate. furthermore, we establish the necessary and sufficient condition for a ring to be a direct sum of its ideals.