Structure theory of graded regular graded self-injective rings and applications

In this paper, we develop structure theory for graded regular graded self-injective rings and apply it in the context of Leavitt path algebras. We show that for a finite graph, graded regular graded self-injective Leavitt path algebras are of graded type I and these are precisely graded $\Sigma$-$V$ Leavitt path algebras.