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    1. Determine (up to isomorphism) all semisimple rings of order 1008. How many of them are
    commutative? (Recall that finite division rings are fields.)

    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
    ring.

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