# Proving a ring-homomorphism using a group-homomorphism

Let f : R → R' be a group homomorphism. Show that the induced map φ : R[x] → R'[x], where φ(anxn + . . . + a0) = f(an)xn + . . . + f(a0), is a ring homomorphism.

I know that φ(0) = f(0) = 0 since f is a group homomorphism. I also know how to show the additive and multiplicative properties for the ring homomorphism, but how can I prove that φ(1) = f(1) = 1?

Also, we're only using commutative rings in my class; I forgot to specify that because I never have to in my work.

I believe it's safe to assume that R, R', R[x], and R'[x] are all rings.

• $\varphi(1) \neq 1$ is possible; consider $f: \mathbb{Z} \rightarrow \mathbb{Z}$ given by $f(r)=2r$. – Matt B Nov 29 '18 at 21:19
• What are $R$ and $R'$? If they are (unital) rings but $f$ is just supposed to be a homomorphism between their additive groups, I see indeed no reason why $f(1_R)$ should be $1_{R'}$. (Edit: And @MattB gives a counterexample.) – Torsten Schoeneberg Nov 29 '18 at 21:19
• In the definition I learned for ring homomorphisms, the multiplicative identity has to be mapped to the multiplicative identity, so φ(1) = 1 – Jon D. Nov 29 '18 at 21:24
• Could you double-check the statement of the question? I would guess it is supposed to say "Let $f: R \to R'$ be a ring homomorphism." – arkeet Nov 29 '18 at 22:34
• Sure, but it is probably a mistake in what is written (so check with your professor?) because it is false as is. – arkeet Nov 29 '18 at 23:13

It should be: "$$f:R\to R'$$ is a ring homomorphism". Otherwise this is not true. Indeed, if $$f$$ is not a ring homomorphism then $$f(ab)\neq f(a)f(b)$$ for some $$a,b\in R$$. It is clear that $$\varphi(ab)\neq\varphi(a)\varphi(b)$$ as well where $$a,b$$ are now treated as polynomials of degree $$0$$. Note that for polynomial $$r$$ of degree $$0$$ we have $$\varphi(r)=f(r)$$.

As an example of such group homomorphism that is not a ring homomorphism but even satisfies $$f(1)=1$$ consider this: let $$R=R'=\mathbb{Z}^2$$ (with pointwise multiplication) and let $$f(x,y)=(x,2x-y)$$. I leave it as an exercise that $$f$$ is a group homomorphism. But it is not a ring homomorphism because

$$f((2,1)\cdot (2,1))=f(4,1)=(4,7)$$ $$f(2,1)\cdot f(2,1)=(2,3)\cdot (2,3)=(4,9)$$

BTW: this example shows that your I also know how to show the additive and multiplicative properties for the ring homomorphism statement cannot be correct (more precisely I'm refering to the "multiplicative" part).

So the assumption "$$f$$ is a group homomorphism" is a mistake (it is not strong enough) and it should be "$$f$$ is a ring homomorphism".

Also note that the identity of $$R[X]$$ is $$1$$ (treated as a polynomial of degree $$0$$). Therefore $$\varphi(1)=1$$ if and only if $$f(1)=1$$. It's quite trivial. More difficult is to show that $$\varphi$$ preserves multiplication if $$f$$ does.

To prove that $$\varphi: R[x]\rightarrow R'[x]$$, is a ring homomorphism you have to show that: $$\varphi(f+g)=\varphi(f)+\varphi(g), \qquad \varphi(fg)=\varphi(f)\varphi(g) \qquad \text{and} \qquad \varphi(1)=1.$$ Note that you have suppose that $$f: R\rightarrow R'$$ is a ring homomorphis.