Abstract
If a’ is arbitrary in f(R) there is an element a ∈ R such that a’ = f(a). Then a’.f(1) = f(a).f(1) = f(a.1) = f(a) = a’ and similarly f(1).a’ = a’ so that f(1) is the identity in f(R). An easy (but not trivial!) counterexample is f : ℤ12 → ℤ12, \( f(\bar x) = \overline {4x} ,\forall \bar x \in {_{12}} \). Here 4̄ is the identity in \(f({_{12}}) = \{ \bar 0,\bar 4,\bar 8\} \) and surely 1̄≠4̄
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© 1998 Springer Science+Business Media Dordrecht
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Cǎlugǎreanu, G., Hamburg, P. (1998). Ring Homomorphisms. In: Exercises in Basic Ring Theory. Kluwer Texts in the Mathematical Sciences, vol 20. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9004-4_21
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DOI: https://doi.org/10.1007/978-94-015-9004-4_21
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4985-8
Online ISBN: 978-94-015-9004-4
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