Prove Theorem 15.1.Reference:

Chapter 15, Problem 1E

(choose chapter or problem)

Prove Theorem 15.1.

Theorem 15.1 Properties of Ring Homomorphisms

Let \(\phi\) be a ring homomorphism from a ring R to a ring S. Let A be a subring of R and let B be an ideal of S.

1. For any \(r \in R\) and any positive integer \(n, \phi(nr) = n \phi(r)\) and \(\phi (r^n) = (\phi(r))^n\).

2. \(\phi(A) = \{f(a) | a \in A\}\)is a subring of S.

3. If A is an ideal and f is onto S, then f(A) is an ideal.

4. \(\phi^{-1}(B) = \{r \in R | \phi(r) \in B\}\) is an ideal of R.

5. If R is commutative, then \(\phi(R)\) is commutative.

6. If R has a unity \(1, S \ne \{0\}\), and \(\phi\) is onto, then \(\phi(1)\) is the unity of S.

7. \(\phi\) 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.