Hi
If by F(R) you mean the set of all maps from to where is a ring, then each element of can be seen as a "sequence" where , and the binary operations you defined over it to obtain a ring are:
addition: ,
multiplication: ,
with as zero and identity element which are obviously idempotent.
When has more than two elements, can you find very simple elements of other than and (but similar...) that are idempotent?