In this paper, the terms, Maximal G- ideal, Primary G-semigroup, prime G-ideal and simple G-semigroup are introduced. It is proved that if S is a G-semigroup containing 0 and identity with the maximal G-ideal M. Then every non zero primary G-ideal is prime as well as maximal if and only if S/M is a 0-simple G-semigroup with either 1) M = (S\M) GaG (S\M) ? {0}, a ? M and G = 0 or 2) M is a 0- simple G-semigroup. Also it is proved that if S is a duo G-semigroup containing 0 and identity with the maximal G-ideal M. Then every non zero primary -ideal is prime as well as maximal if and only if S is one of the following types 1) S = G ? M where G is the G-group of units and M = {aγg : g ? G, aγa = 0, a ? M, γ ? G } ? {0}. 2) S is the union of two G-semigroups with 0-adjoined. Also it is proved that if S is a commutative G-semigroup with 0 and identity and with the maximal G-ideal M. Suppose that every non zero primary G-ideal is prime or every nonzero G-ideal is prime. Then S satisfies either one of the following conditions 1) S = G?M, where G is the G-group of units in S and M = (a GG) ? {0}, a ? M and a Ga = 0 2) (MG) n-1 M = M for every positive integer n. Furthermore if S has maximum condition on G-ideals then for every m ? M, we have m ? M Ge, e being a proper idempotent and also proved that if S is a quasi commutative Noertherian G-semigroup containing identity. Suppose every primary G-ideal in S is prime. Then the following are equivalent 1) S is cancellative. 2) S has no proper G-idempotents. 3) S is a G-group.