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Let $\phi\colon R \to R'$ be a ring homomorphism. Prove that if $R$ is a field then either $\phi$ is an isomorphism or $\phi(r) = 0$ for all $r \in R$.

I am stuck on this problem and don't know where to begin. I feel like I'm very weak in writing proofs.

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    **Hint** $\ $ There are only a couple possibilities for the ideal ker $\phi$ given that R is a field.2011-05-03
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    I'm sorry I don't know what your hint refers to. My professor skips around and sometimes goes back whenever a student reminds him that he hasn't proven something that he thinks is obvious2011-05-03
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    You must be using "isomorphism" to mean "one-to-one homorphism"; otherwise, $\mathbb{R}\hookrightarrow\mathbb{C}$ would be a counterexample (not to mention $\mathbb{R}\hookrightarrow \mathbb{R}\times\mathbb{Z}$).2011-05-03
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    As stated, this is *not* true: Consider for instance the inclusion of $\mathbb Q \to \mathbb R$. So, you need some other assumption, like $\phi$ is surjective. Also, I assume you mean $\phi$ is an isomorphism, not $R$.2011-05-03
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    @lhf: Some books use "isomorphism" to mean "one-to-one", and "isomorphism onto" to mean bijective morphism. E.g., Herstein's *Topics in Algebra*.2011-05-03
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    @Arturo Yes two Rings have an isomorphism doesnt mean that they are isomorphic. Surjection is needed to make the "isomorphic". I thought it is a standard notation. Is it not?2011-05-04
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    @Dinesh: No, it's not, but as I mentioned, it's used in some books. The more or less standard notation is to use "isomorphism" only for bijective/invertible morphisms. Many people use "monomorphism" for one-to-one homomorphisms, though in the context of rings that is technically incorrect (if you are using 'monomorphism' in the categorical sense). But all of this is neither here nor there: it's not standard, but it is not too rare either (as you could tell by the fact that I recognized it and could even cite a good book that follows it).2011-05-04
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    @Arturo Thanks for the info!2011-05-08
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    This business of isomorphisms not necessarily being surjective *was* standard once upon a time (check the publication date of Herstein's book!), and among the French the practice was carried on a little longer, but I haven't seen a reputable source use this convention in the last 20 years.2011-05-13

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