Assignment:
Q1) 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.
Q2) 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.
Q3) 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).
Provide complete and step by step solution for the question and show calculations and use formulas.