My question is simple, though it proves to be much more difficult than it sounds. Suppose I want to find a binary operation to add extra structure to a multiplicative group (so it becomes a ring). This binary operation must be associative, and also distributive over multiplication by definition.
My question is whether or not there exists a binary operation that is both associative and distributive over multiplication. Keep in mind that I am just an amateur mathematician, not an advanced ring theorist, so it would be kind for you to keep things simple.