How many principal prime ideals are there in a polynomial ring?

초록

Let R be a commutative ring with identity, X be an indeterminate over R, R[X] be the polynomial ring over R, R[[X]] be the power series ring over R, and Q be a principal prime ideal of R[X] with (Q ∩ R)[X] ⊈ Q. It is well known that if R is an integral domain, then R[X]Q is a DVR and R[X] has infinitely many such principal prime ideals. In this paper, among other things, we show that (i) RQ∩R is a field, (ii) R[X]Q is a DVR, but (iii) there is a ring R such that R[X] has no principal prime ideal. We also study the maximal ideals of R[[X]] that are principal. © 2026 World Scientific Publishing Company.

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