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Problem 37 (Properties Of Ring Homomorphisms)

Let $\phi:R\to S$ be a ring homomorphism. Prove the following properties.

  1. For any $r\in R$ and any positive integer $n$, $\phi(n\cdot r) = n\phi(r)$ and $\phi(r^n) = (\phi(r))^n$.
  2. If $A$ is a subring of $R$, then $\phi (A) = \{\phi(a)|a\in A\}$ is a subring of $S$.
  3. If $A$ is an ideal, then $\phi(A)$ is an ideal of $\phi(R)$.
  4. If $B$ is an ideal of $S$, then $\phi^{-1}(B) = \{r\in R|\phi(r)\in B\}$ is an ideal of $R$.
  5. If $R$ is commutative, then $\phi(R)$ is commutative.
  6. If $R$ has a unity 1, if $S\neq \{0\}$, and if $\phi$ is onto, then $\phi(1)$ is the unity of $S$.
  7. We know $\phi$ is an isomorphism if and only if $\phi$ is onto and $\ker\phi = \{r\in R|\phi(r)=0\} = \{0\}$.
  8. If $\phi$ is an isomorphism from $R$ onto $S$, then $\phi^{-1}$ is an isomorphism from $S$ onto $R$.
Let me know which parts you are ready to prove in class. We'll split this one up and have several presenters.
Note: If you decide to type up this for your problem, you can pick three of them and prove just them.

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