# subset

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.

https://brainmass.com/math/integrals/isomorphism-subjective-subsets-239522

#### Solution Summary

This solution is comprised of a detailed explanation to solve subset question.