In attempting to prove an algebraic identity in physics, I found that my proof was simplified in appearance if i used an operation a@b=a + b - ab By flipping through a book on ring theory, I found that this is called a "circle composition operation", and it is used in one definition of the Jacobson radical. I'm not a mathematician and my highest formal class in abstract algebra is some basic group theory. What is special about this operation? Why is it interesting? What is special about the Jacobson radical? Why is it interesting?