A module M is said to be extending (G-extending) if for each submodule X of M there exists a direct summand D of M such that X is essential in D (X boolean AND D is essential in both X and D). It is known that for a nonsingular module the concepts of G-extending and extending coincide. However, in the not nonsingular case, they are distinct. In this article, we obtain a characterization of the right G-extending generalized triangular matrix rings. This result and its corollaries improve and generalize the existing results on right extending generalized triangular matrix rings. It is well known that the ring of n-by-n triangular matrices over a right selfinjective ring is not, in general, right extending. One application of our characterization shows that such rings are right G-extending. Connections to Operator Theory and a characterization of the class of right extending right SI-rings are also obtained. Examples are given to illustrate and delimit the theory.