Unformatted text preview: c = b x = ( x 1 , x 2 , . . . ) : x k ∈ R and lim k →∞ x k exists B , c = b x = ( x 1 , x 2 , . . . ) : x k ∈ R and lim k →∞ x k = 0 B . You may assume without proof that these are both commutative rings under the following operations: if x = ( x 1 , x 2 , . . . ) and y = ( y 1 , y 2 , . . . ) then x + y = ( x 1 + y 1 , x 2 + y 2 , . . . ) and xy = ( x 1 y 1 , x 2 y 2 , . . . ) . (a) Determine (with proof) whether c or c has a multiplicative identity. (b) Show directly that c is a (twosided) ideal in c . (c) Use the First Homomorphism Theorem to show that c/c ∼ = R . (d) Given x ∈ c , give an explicit description of the coset x + c . 1...
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 Spring '10
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 Math, Algebra, Addition, xy, lim xk

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