I am thinking that a good place to start is to figure out what are the neutral elements with respect to the operations ! and @ in T: For what element x in the ring R will x ! a = a for all a? For what element y in R will y@a = a for all a in R? (Notice that both ! and @ are commutative, so it suffices to check that x and y work from the left).
Then you could find inverses with respect to !, i.e. for a given a, what element b satisfied that a ! b = x, where x is the neutral element from before?