On the (graded) Jacobson radical of graded alternative rings with an application to the power graphs of power-associative magmas

Let S be a cancellative semigroup with the set E(S) of all of its idempotent elements, and let R=⨁s∈SRs be a not necessarily associative S-graded ring. We provide a source of examples of graded alternative rings with a graded version of Cayley–Dickson algebras, and we establish a graded version of Kleinfeld’s classification of all simple alternative rings. The main focus is on defining and studying the structure of the graded Jacobson radical of R, in particular when R is alternative, and its relation to the classical Jacobson radical. If R is alternative, the set HR=⋃s∈SRs is a power-associative magma with respect to the ring multiplication. We apply the obtained results to address a problem raised in [Abawajy, J., Kelarev, A., Chowdhury, M.: Power graphs: a survey. Electron. J. Graph Theory Appl. 1(2), 125–147 (2013)], by characterizing, under certain assumptions, the connectedness of the power graph G(HR) in terms of the connectedness of subgraphs induced by Re, where e runs through E(S).