2. A ring R is Boolean if x2 = x for all x R. In a Boolean ring R, show that
a. 2x = 0 for all x R.
b. R is commutative.
c. Every prime ideal P is maximal and R/P is a field with 2 elements.
d. Every finitely generated ideal in R is principal.
3. Let X be a topological space which is compact, Hausdorff and totally disconnected (i.e.,
given any x, y X, there is a set U X which is both open and closed (clopen) such that x U, y U). The space X is called a Boolean space. Let R be the set of all clopen subsets of X.
For U, V R, define U + V = (U V ) (U ∩ V ), the symmetric difference of
U and V; and define U ∙ V = U ∩ V. Show that these operations make R into a Boolean
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