ass7 - ,x ) be the ideal generated by 2 and x . (i) Prove...

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ASSIGNMENT 7 - MATH235, FALL 2007 Submit by 16:00, Monday, November 5 1. The ring of real quaternions H . Let i,j,k be formal symbols and H = { a + bi + cj + dk : a,b,c,d R } . Addition on H is defined by ( a + bi + cj + dk ) + ( a 0 + b 0 i + c 0 j + d 0 k ) = ( a + a 0 ) + ( b + b 0 ) i + ( c + c 0 ) j + ( d + d 0 ) k. Multiplication is determined by defining i 2 = j 2 = - 1 , ij = - ji = k, (and one extend this to a product rule by linearity). (1) Prove that the map H ‰± z 1 z 2 - z 2 z 1 : z 1 ,z 2 C ² , taking a + bi + cj + dk to the matrix ± a + bi c + di - c + di a - bi is bijective and satisfies: f ( x + y ) = f ( x ) + f ( y ) and f ( i ) 2 = f ( j ) 2 = - I 2 ,f ( i ) f ( j ) = f ( k ) = - f ( j ) f ( i ). (2) Use the previous question to conclude that H is a ring. (3) Prove that H is a non-commutative division ring. 2. (1) Prove that there is no ring homomorphism Z 5 Z . (2) Prove that there is no ring homomorphism Z 5 Z 7 . 3. (1) Let R be any commutative ring and let a 1 ,...,a n be elements of R . We define ( a 1 ,...,a n ) to be the set { r 1 a 1 + ··· + r n a n : i r i R } . Prove that ( a 1 ,...,a n ) is an ideal of R . We call it the ideal generated by a 1 ,...,a n . (2) Now apply that to the case where R = Z [ x ] (polynomials with integer coefficients). Let (2
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Unformatted text preview: ,x ) be the ideal generated by 2 and x . (i) Prove that the ideal (2 ,x ) is not principal and conclude that Z [ x ] is not a principal ideal ring. (ii) Find a homomorphism f : R Z 2 such that (2 ,x ) = Ker( f ) . 4. Prove that no two of the following rings are isomorphic: (1) R R R R (with addition and multiplication given coordinate by coordinate); (2) M 2 ( R ); (3) The ring H of real quaternions. 5. Let f : R S be a ring homomorphism. (1) Let J C S be an ideal. Prove that f-1 ( J ) (equal by denition to { r R : f ( r ) J } ) is an ideal of R . (2) Prove that if f is surjective and I C R is an ideal then f ( I ) is an ideal (where f ( I ) = { f ( i ) : i I } ). (3) Show, by example, that if f is not surjective the assertion in (2) need not hold....
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