$R[X]$ is a principal ideal ring if and only if $R\cong R_{1}\oplus R_{2}\oplus\cdots \oplus R_{n}$,where each $R_{i}$ is a field.

Asked Nov 28 '16 at 07:21

Active Nov 28 '16 at 10:16

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I have read from the Gallian's abstract algebra textbook the following result

$R[X]$ is a principal ideal ring if and only if $R\cong R_{1}\oplus R_{2}\oplus\cdots \oplus R_{n}$, where each $R_{i}$ is a field.

which the book said it was proved by undergraduates. And I try to prove it on my own, but it resists all my attempts. I try to seek the original paper on Internet, but I haven't got a clue. I am eager to know how to prove it. Any help or hint will be appreciated, thank you so much!

edited Nov 28 '16 at 10:16

user26857

52,094

asked Nov 28 '16 at 07:21

W.Leywon

3

Related: https://math.stackexchange.com/questions/91587/ring-of-polynomials-is-a-principal-ideal-ring-implies-coefficient-ring-is-a-fiel, https://math.stackexchange.com/questions/361258 – Watson Nov 28 '16 at 07:44
Thank you,although I can't fully understand the profound conclusions your link links.I mean,is there an elementary proof of that? Thanks. – W.Leywon Nov 28 '16 at 09:06
Principal ideal RING,not Domain. – W.Leywon Nov 28 '16 at 14:56

$R[X]$ is a principal ideal ring if and only if $R\cong R_{1}\oplus R_{2}\oplus\cdots \oplus R_{n}$,where each $R_{i}$ is a field.

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