On graded division rings

Abstract

Given an associative ring with identity $R$ and a ring homomorphism $f:R\to D$ 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 $\Gamma$ we adapt the theory of Cohn to determine the different homomorphisms of graded rings from $R$ to $\Gamma$-graded division rings.