Boolean Rings, Homomorphisms, Isomorphisms and Idempotents
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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).
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Solution Summary
Boolean Rings, Homomorphisms, Isomorphisms and Idempotents are investigated. Trivial idempotent elements are discussed.
Solution Preview
1) The Boolean ring of P(X) has operations + = symmetric difference,
and * = intersection. fix i and thus f_i.
Now if A,B are in R, then
1 = f(A+B) iff i is in A+B iff i is in AB or i is in BA
iff i is in A and not in B OR i is in B and not in A.
iff f_i(A)=1 and f_i(B)=0 OR f_i(B)=1 and f_i(A) = 0
if f_i(A) + f_i(B) = 1 (in Z_2)
And there are only two possibilities for f_i(A+B), so we always ...
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