In principal ideal domain

Question

Hence that $K$ is principal ideal domain. Prove that every ring $R$ is isomorphic with $K$ is principal ideal domain.

Answer 1

Give $I$ is a ideal of $R$. We must prove that $I$ is principal ideal.

By $R$ is isomorphic with $K$ that exist $f: R \rightarrow K$ is isomorphic.

When $f(I) \in K$ and $K$ is P.I.D then exist $a\in K$ that $f(I) = (a) = aK$.

By $f$ is isomorphic then exist reverse mapping $f^{-1}: K \rightarrow R$ is isomorphic. Thus, $I = f^{-1}\left[f(I) \right] = f^{-1}(aK) = f^{-1}(a)f^{-1}(K)=f^{-1}(a)R$

Infer I is principal domain.

So $R$ is P.I.D