Let ϕ : F → R be a ring homomorphism from a field F into a ring R. Prove that if ϕ ( a ) = 0 for some nonzero a ∈ F , then ϕ ( x ) = 0 for all x ∈ F and the factor ring R / ker ⁡ ( ϕ ) is trivial.

Let ϕ : F → R be a ring homomorphism from a field F into a ring R. Prove that if ϕ ( a ) = 0 for some nonzero a ∈ F , then ϕ ( x ) = 0 for all x ∈ F and the factor ring R / ker ⁡ ( ϕ ) is trivial.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.3: The Field Of Quotients Of An Integral Domain

Problem 8E

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