# Prove some properties of ring homomorphism

abstract-algebraring-homomorphismring-theory

Let $$R,S$$ be rings and $$\varphi : R\to S$$ be a ring homomorphism. Verify that

1. $$\varphi(na) = n\varphi(a)$$ for all $$n\in\mathbb Z$$ and $$a\in R$$.
2. $$\varphi(a^n) = (\varphi(a))^n$$ for all $$n\in\mathbb Z^+$$ and all $$a\in R$$.
3. If $$A$$ is a subring of $$R$$, then $$\varphi(A) = \{\varphi(a):a\in A\}$$ is a subring of $$S$$.

For (1) I have the following:
$$\begin{split} \varphi(na) &= \varphi((n-1)a+a) \\ &= \varphi((n-1)a)+\varphi(a) \\ &= (n-1)\varphi(a)+\varphi(a)\\ &= n\varphi(a). \end{split}$$

For (2): For $$n=1$$, we have $$\varphi(a) = \varphi(a)$$. Suppose for some $$n\in\mathbb Z^+$$ that $$\varphi(a^n)=(\varphi(a))^n$$. Observe:
$$\begin{split} \varphi(a^n) & =(\varphi(a))^n \\ \varphi(a)\cdot \varphi(a^n) & =\varphi(a)\cdot (\varphi(a))^n \\ \varphi(a \cdot a^n) & =(\varphi(a))^{n+1} \\ \varphi( a^{n+1}) & =(\varphi(a))^{n+1} \\ \end{split}$$

For (3): I verify for $$a,b\in\varphi(A)$$ then $$a-b\in\varphi(A)$$ and $$a\cdot b\in \varphi(A)$$ (for brevity).

• Well, $$0\in A \implies 0 \in \varphi(A)$$
• For $$a,b \in \varphi(A)$$ we have $$\varphi(a) – \varphi(b) = \varphi(a-b)$$, thus, verified, because $$a-b\in A$$.
• For $$a,b \in \varphi(A)$$, we have $$\varphi(a) \cdot \varphi(b) = \varphi (a\cdot b)$$, thus, verified, because $$a\cdot b \in A$$.

Did I do these right? Feedback appreciated!

For the first property, use induction to show that $$\phi(na) = n\phi(a)$$ for all $$a$$ and $$n\geq 0$$. This is actually what you have done.
For negative $$n$$ one must be careful. For $$n>0$$ put $$(-n)a = -(na)$$, which is the additive inverse of $$na$$. This part is then true simply by definition.