Polynomial Rings : Prove that f (x) is a zero - divisor in A[x] IF AND ONLY IF (iff) 7c ! A, c ! 0 " cf (x) = 0.
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Let A be a commutative ring with identity 1, and let A[x] be the ring of polynomials with coefficients in A.
Let f (x) = a0+ a1x + ... + anxn! A[x] .
Prove that f (x) is a zero - divisor in A[x] IF AND ONLY IF (iff) 7c ! A, c ! 0 " cf (x) = 0.
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Commutative rings and a zero-divisor are investigated. The solution is detailed and well presented.
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Recall: let R is a ring. Let a 0 R. Then we say a is a zero divisor if there exist b 0 R such that a.b = 0
Let f(x) = a0 + a1x + ...+ anxn. If a0 is a zero divisor of A, then it is a zero divisor of A[x] such that a0b0 =0 for some b0 A. Let g(x) = b0 + b1x+ ... + bmxm A[x]. Then we have
a0.g(x) = ...
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