By David R. Finston and Patrick J. Morandi

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11. Let T be a set, and let R be the set of all subsets of T . We have two binary operations on R, namely union and intersection. However, R does not form a ring under these two operations. The identity element for union is the empty set since A[ = [A = A for any A 2 T: Similarly R serves as the identity element for intersection. On the other hand one can check that neither operation allows for inverses: given a nonempty subset A of R, what subset B can satisfy A[B = ? And if A is a proper subset of T what subset B can eatisfy A\ B = R?

B) Prove or disprove: if a; b 2 R, then (ab)n = an bn . If this is not always true, determine under what circumstances it is true. 20. De…ne a relation between elements of a commutative ring R by a b if there is a unit u 2 R with b = au. Prove that this relation is an equivalence relation. Determine the equivalence classes of this relation if R = Z and if R = Z12 . 21. Is subtraction on Z an associative operation? Is it commutative? Why or why not? 22. Let R be a ring. Suppose that S is a subset of R containing 1 satisfying (i) if a; b 2 S, then a b 2 S, and (ii) if a; b 2 S, then ab 2 S.

We saw that the ring properties hold for Zn in the previous chapter. We also saw that multiplication is commutative, so Zn is a commutative ring. 46 CHAPTER 3. 9. Let R be the set of all continuous (real valued) functions de…ned on the interval [0; 1]. Recall that these function operations are de…ned pointwise. That is, if f and g are functions, then f + g, f g, and f g are de…ned by (f + g)(x) = f (x) + g(x); (f g)(x) = f (x) g(x); (f g)(x) = f (x)g(x): In calculus one shows that the sum, di¤erence, and product of continuous functions is again continuous.