On the ring structure of one-value graph magma algebras

This paper investigates the ring structure of one-value graph magma algebras, a class of monoid algebras induced by simple directed graphs. We explore the general structure of these algebras, focusing on their Jacobson radical, left and right ideals, and the conditions under which they are semiperfect. A key result is the decomposition of any semiperfect graph magma algebra into a direct sum of a semisimple ring and a ring isomorphic to a graph magma algebra induced by a graph without isolated idempotent vertices. This decomposition simplifies the analysis by allowing us to restrict our attention to algebras without isolated idempotent vertices. We also examine the socle and singular ideals of these algebras, establish conditions for injectivity and semisimplicity of modules, and identify when these algebras are right Kasch rings.