If $P(x)$ is the ring of all subsets of a given set $x,$ prove that

a$)$ the collection of all finite subsets of $x$ forms an ideal of $P(x)$;

b$)$ for each subset Ysubex,$P(Y)$ and $P(x-Y)$ are both principal ideals of $P(x),$

with $P(x)=P(Y)o+P(x-Y).$