I think that to find all homomorphisms requires the natures of both the rings .
However,could I focus on the idempotents(use the muiltipication nature),or could I use additions natures(such as $(a,b)+$any element$=$the element I have chosen i.e.$(a,b)$ is the addtion identity element)
but I always found troubles:missing some potential homo.
I consider using ring natures is a nice way ,but are there any other points to pay attention?or consider its group?or a formal rout to follow!?