In this paper, we establish several new results on commutative G-graded rings where G is a totally ordered abelian group. McCoy’s theorem and Armendariz’ theorem are classical results in the theory of polynomial rings. We generalize both of these celebrated theorems to the more general setting of G-graded rings and simultaneously to the setting of ideals rather than to that of elements. Next, we give a complete characterization of invertible elements in G-graded rings. We generalize Bergman’s famous theorem (which asserts that the Jacobson radical of a Z-graded ring is a graded ideal) to the setting of G-graded rings and then proceed to give a natural and quite elementary proof of it. This generalization allows us to show that an abelian group is a totally ordered group if and only if the Jacobson radical of every ring graded by that group is a graded ideal, or equivalently, nonzero idempotents of every ring graded by that group are homogeneous of degree zero. Finally, some topological aspects of graded prime ideals are investigated.