Derivations of special subrings of matrix rings and regular graphs (Q791616)

Let \(I_ n=\{1,2,...,n\}\) and let \(\rho\) be a reflexive transitive relation on \(I_ n\). Let \(G\) be an Abelian group (written additively). A map \(f:\rho \to G\) is called transitive if \(f(i,j)+f(j,k)=f(i,k)\) (for \(i\rho j,j\rho k)\) and trivial if \(f(i,j)=\sigma(i)-\sigma(j)\) for some map \(\sigma:I_ n\to G.\) The author calls \(\rho\) regular on \(G\) if every transitive map \(f:\rho\to G\) is trivial. He relates this concept of regularity to the concept of a regular graph. Let \(R\) be a ring with identity and let \(M_ n(R)\) be the ring of \(n\times n\) matrices over \(R\). Given a relation \(\rho\) on \(I_ n\), let \[ P=\{A=(a_{ij})\in M_ n(R)\mid a_{ij}=0\;\text{if}\;(i,j)\not\in \rho \}. \] The author calls \(P\) the special subring with the relation \(\rho\). The main result proves that the following two conditions are equivalent (i) every \(R\)-derivation of \(P\) is inner, (ii) \(\rho\) is regular on \(Z(R)\). He also shows that these conditions are equivalent to a cohomological condition which is to technical to state here. The paper concludes with some interesting graph-theoretic applications.