Boolean Rings, Homomorphisms, Isomorphisms and Idempotents

1)Let X={1,2,...,n}and let R be the Boolean ring of all subsets of X.
Define f_i:R->Z_2 by f_i(a)=[1] iff i is in a.Show each f_i is a
homomorphism and thus f=(f_1,...,f_n):R->Z_2*Z_2*...*Z_2 is a ring
homomorphism.Show f is an isomorphism.

2)If T is any ring,an element e of T is called an idempotent provided
e^2=e.The elements 0 and 1 are idempotents called the trivial idempotents.
Suppose T is a commutative ring and e in T is an idempotent with 0/=e/=1
(/=:is not equal to).Let R=eT and S=(1-e)T.Show each of the ideals R and S
is a ring with with identity,and f:T->R*S defined by f(t)=(et,(1-e)t) is
a ring isomorphism.

3)Use the result from 2) to show that any finite Boolean ring is
isomorphic to Z_2*Z_2*...*Z_2, and thus also to the Boolean ring of
subsets of 1).