On graded division rings

Given an associative ring with identity R and a ring homomorphism f R→ D to a division ring D , the singular kernel of f is the set of square matrices of all sizes over R , which, on applying f , yield singular matrices over D . Paul M. Cohn characterized the sets of square matrices that can arise as singular kernels and showed that, up to isomorphism, the singular kernels characterize the different homomorphisms from R to division rings. In this work, we show that this characterization can be implemented in the context of graded rings. More precisely, given a ring R graded by a group Γ we adapt the theory of Cohn to determine the different homomorphisms of graded rings from R to Γ -graded division rings.