If $R[x]$ is PID then $R$ is a field

Asked Oct 25 '23 at 22:28

Active Oct 25 '23 at 23:59

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I am solving a question in Ring and Modules. The question is as follows: $R$ is commutative ring with identity.

If $R[x]$ is PID then $R$ is a field.

I have solved the problem by considering the quotient $R[x]/(x) $.

But I found that in the exercise it is instructed to solve the problem considering.... "$R$ is PID iff every submodule of $R$ is free".

I have no idea. Can you please help me. Any hint will be welcomed. Thanks.

edited Oct 25 '23 at 22:31

asked Oct 25 '23 at 22:28

Sayantan73636

1

That is merely a restatement of the definition of a PID, I don't think that this leads anywhere. Just take the usual proof. – Martin Brandenburg Oct 25 '23 at 22:53

0 Answers0