midterm2review

# midterm2review - of R : C ( b ) = { a R : ab = ba } . 8....

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Midterm 2 Review (not collected) 1. Let p be a positive prime number. Prove that if [ a ] 6 = [0], then [ a ] is a unit in Z p . 2. Is M ( R ) an integral domain? (Recall that M ( R ) is our textbook’s way to denote the ring of 2 × 2 matrices with real entries.) 3. Deﬁne f : R M ( R ) via f ( a ) = ± a 0 0 - a ² Is f a homomorphism? 4. Let m Z , and assume that m 1. Prove that the set m Z = { a Z : m | a } is a subring of Z . 5. Let R = { ( m,n ) : m,n Z } . Deﬁne ( m,n ) + ( p,q ) = ( mq + np,nq ) and ( m,n ) * ( p,q ) = ( mp,nq ). Is ( R, + , * ) a ring? 6. Suppose that D is an integral domain, and let a D . Prove that the equation x 2 = a 2 has at most two solutions in D . Give an example to show that this statement is false if D is only assumed to be a ring. 7. Let R be a ring and let b R . Prove that the following set is a subring
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Unformatted text preview: of R : C ( b ) = { a R : ab = ba } . 8. Give an example of a ring which is an integral domain but which is not a eld. 9. Prove that if f : R S is a homomorphism, then the following set is a subring of R : K = { a R : f ( a ) = 0 S } . 10. (Bonus) Let p be a prime. Prove that ( p-1)! -1(mod p ). (This result is known as Wilsons theorem.) Hint: Prove that [ a ] = [ a ]-1 if and only if [ a ] = [-1] or [ a ] = [1]. (Compare with problem number 6 above.) Then use the fact that every nonzero element of Z p is a unit. 1...
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## This note was uploaded on 09/08/2011 for the course MTH 310 taught by Professor R.bell during the Spring '10 term at Michigan State University.

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