Unformatted text preview: Z 12 / ([ 4 ]) (which in this case is the same as Z 12 / h [ 4 ] i ) as 0 = { [ ] , [ 4 ] , [ 8 ] } , 1 = { [ 1 ] , [ 5 ] , [ 9 ] } , 2 = { [ 2 ] , [ 6 ] , [ 10 ] } , 3 = { [ 3 ] , [ 7 ] , [ 11 ] } . Then the Cayley tables will be ⊕ 1 2 3 1 2 3 1 1 2 3 2 2 3 1 3 3 1 2 and ± 1 2 3 1 1 2 3 2 2 2 3 3 2 1 2. Problem 39.3 on page 183. Prove that if R is commutative and I is an ideal of R , then R / I is commutative. The general element of R / I is of the form I + a where a ∈ R . A ring is commutative means that the operation of multiplication is commutative. Therefore we need to prove that ( I + a )( I + b ) = ( I + b )( I + a ) for all a , b ∈ R , in other words I + ab = I + ba for all a , b ∈ R . Since R is commutative, we have ab = ba and the result follows....
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 Fall '08
 PARRY
 Math, Algebra, Addition, Fundamental theorem on homomorphisms, Fundamental Homomorphism Theorem, Ker θ

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