# Ring of Polynomials is a Principal Ideal Ring implies Coefficient Ring is a Field?

I read this proof that if $D$ is an integral domain and $D[X]$ is a principal ideal domain, then $D$ is a field.

My question is if the requirements can be relaxed a bit, namely:

Is it true that if $D$ is a commutative unitary ring and $D[x]$ is a principal ideal ring (this allows zero-divisors), then $D$ is a field?

I would be very pleased if anyone could give me a counter-example or could sketch a proof, certainly the linked proof would completely break down in this case as one could not use the properties of degree.

-
Interesting question! Clearly, if $A$ is a (commutative unitary) noetherian ring such that every ideal of $A[x]$ is principal, then $A$ is artinian. – Andrea Dec 14 '11 at 22:30
For the record (for those who come here and may have difficulty extracting this from the given answer), the answer to the question is no, "$D[X]$ a principal ring" is equivalent to "$D$ is a finite product of fields", which of course is not a field if there are at least two factors (or none at all). See also my question which is marked as duplicate of this one, but has more answers. – Marc van Leeuwen Apr 15 '13 at 5:24

More generally here is a semigroup version (from my old sci.math post). Please feel quite welcome to edit it (I don't have time now to TeX it).

THEOREM $\ \$ TFAE for a semigroup ring R[S], with unitary ring R, and nonzero torsion-free cancellative monoid S.

1) $\$ R[S] is a PIR (Principal Ideal Ring)
2) $\$ R[S] is a general ZPI-ring (i.e. a Dedekind ring, see below)
3) $\$ R[S] is a multiplication ring (i.e. $\rm\ I \supset\ J \Rightarrow\ I\ |\ J\$ for ideals $\rm\:I,J\:$)
4) $\$ R is a finite direct sum of fields, and S is isomorphic to $\mathbb Z$ or $\mathbb N$

A general ZPI-ring is a ring theoretic analog of a Dedekind domain i.e. a ring where every ideal is a finite product of prime ideals. A unitary ring R is a general ZPI-ring $\iff$ R is a finite direct sum of Dedekind domains and special primary rings (aka SPIR = special PIR) i.e. local PIRs with nilpotent max ideals. ZPI comes from the German phrase "Zerlegung in Primideale" = factorization in prime ideals. The classical results on Dedekind domains were extended to rings with zero divisors by S. Mori circa 1940, then later by K. Asano and, more recently, by R. Gilmer. See Gilmer's book "Commutative Semigroup Rings" sections 18 (and section 13 for the domain case).

49#5213 20M25 (13F05)
Gilmer, Robert; Parker, Tom. Semigroup rings as Prufer rings.
Duke Math. J. 41 (1974), 219--230.

Let RS be the semigroup ring of a torsion-free cancellative abelian semigroup S with zero over a commutative ring R with identity. The semigroup operation on S is written as addition. Such rings RS may be essentially thought of as generalizations of polynomial rings. The authors seek conditions on R and S under which the semigroup ring RS will have a given ring-theoretic property. Some necessary and sufficient conditions are found for the ring RS to fall into one of four classes of rings: Prufer rings, Bezout rings, almost Dedekind rings, and general ZPI-rings. The investigations are closely related to (but independent of) another paper of the authors [Michigan Math. J. 21 (1974), 65--86; MR 49#7381]. In Sections 2 and 3, the case of Prufer rings is considered. A Prufer ring is a commutative ring R with identity such that each finitely generated regular ideal of R is invertible. In order to state the main result of these two sections, we need some more definitions: Let Z, [Q] be the additive group of integers [rationals], and let Z_0, [Q_0] be the additive semigroup of nonnegative integers [nonnegative rationals]. Semigroups of the form G /\ Q_0, where G is a subgroup of Q containing Z , are called Prufer sub-semigroups of Q_0. A commutative ring with identity in which each finitely generated ideal is principal is a Bezout ring. It is shown that if RS is a Prufer ring then R is (von Neumann) regular. Further, the authors prove the equivalence of the following three conditions: (1) RS is a Prufer ring; (2) RS is a Bezout ring; (3) R is a regular ring, and to within isomorphism S is either a Prufer subsemigroup of Q_0 or a subgroup of Q containing Z . In Section 4 the authors deal with almost Dedekind rings (AD-rings). Following M. D. Larsen [J. Reine Angew. Math. 245 (1970), 119--123; MR 42#7662], an AD-ring is a Prufer ring in which regular prime ideals are maximal and not idempotent. Some necessary and sufficient conditions for RS to be an AD-ring are found in this section (Theorems 4.1 and 4.2). In the final section (5), the notion of a general ZPI-ring is introduced. These are commutative rings with identity in which each ideal is a finite product of prime ideals. The following result (Theorem 5.1) is now established. The semigroup ring RS is a general ZPI-ring if and only if R is a finite direct sum of fields and S is isomorphic to Z_0 or to Z . Reviewed by Uno Kaljulaid

82d:13019 13F20 (13F05)
Hardy, Bonnie R.; Shores, Thomas S. Arithmetical semigroup rings.
Canad. J. Math. 32 (1980), no. 6, 1361--1371.

Throughout this paper, the ring R and the semigroup S are commutative with identity; moreover, it is assumed that S is cancellative. An arithmetical ring is a ring for which the ideals form a distributive lattice and a ZPI-ring is one in which every ideal is a product of prime ideals.

The authors determine necessary and sufficient conditions on R and S that the semigroup ring R[S] be arithmetical [respectively, semihereditary, a ZPI-ring, a PIR (principal ideal ring)]. The main result is Theorem 3.6: Let R and S be as above and G the group of quotients of S . Let \rho be a congruence defined on S by x\rho y if and only if x=y+f for some f in(S/\ tG) . Then R[S] is arithmetical if and only if one of the following holds: (1) the torsion subgroup tG of G is a proper subsemigroup of S , R[tG] is regular and the semigroup S/\rho of congruence classes of \rho is isomorphic to an additive subgroup of Q or the positive cone of such a group, (2) R is arithmetical and S=G is a torsion group such that if its p -primary component G_p != 0 for some prime p = Char}(R/M), where M is a maximal ideal of R , then G_p is cyclic or quasicyclic and R_M is a field. Two other theorems, 4.1 and 4.2, provide characterizations of R[S] that are ZPI-rings and PIRs. This paper is closely related to the paper by R. Gilmer and T. Parker [Duke Math. J. 41 (1974), 219--230; MR 49#5213], particularly the following results (Corollary 3.1 and Corollary 5.1): If R and S are as above and moreover S is torsion-free, then (a) R[S] is a Bezout ring if and only if R[S] is a Prufer ring if and only if R is a (von Neumann) regular ring and S is isomorphic to an additive subgroup of Q or the positive cone of such a subgroup (the authors point out that each of the above statements is also equivalent to another statement " R[S] is arithmetical"), and (b) R[S] is a ZPI-ring if and only if R[S] is a PIR. Applying their theorems, the authors give examples to show that the above results of Gilmer and Parker are no longer true if the condition "S is torsion-free" is dropped.
Reviewed by Chin-Pi Lu

40 #1380 13.50
Wood, Craig A. On general Z.P.I.-rings.
Pacific J. Math. 30 1969 837--846.

A general Z.P.I.-ring is a commutative ring R each ideal of which is a finite product of prime ideals. Consider the cases (A) R has an identity, (B) R has no identity, but has at least one proper prime ideal, and (C) R has neither identity nor proper prime ideal. In each case the author gives firstly a structure theorem for general Z.P.I.-rings and secondly criteria for R to be a general Z.P.I.-ring. The structure theorems, which have been given in a less clear form by S. Mori [J. Sci. Hiroshima Univ. Ser. A 10 (1940), 117--136; MR 2, 121], are as follows. R is a general Z.P.I.-ring if and only if R is a finite direct sum of Dedekind domains and special P.I.R.'s in case (A), R = F (+) T in case (B) and R = T in case (C), where F is a field and T is a ring without identity and without non-zero ideals other than powers of T .
Reviewed by D. Kirby

13,313e 09.1X
Asano, Keizo. Uber kommutative Ringe, in denen jedes Ideal als Produkt von Primidealen darstellbar ist. (German)
J. Math. Soc. Japan 3, (1951). 82--90.

Let a commutative ring R with identity element be called a Dedekind ring if it is the direct sum of a finite number of Dedekind integral domains and of rings having a nilpotent, principal, maximal ideal. Various conditions on R are proved equivalent to its being Dedekind, among them the following: (1) Every ideal in R is a product of prime ideals; (2) the zero ideal is a product of prime ideals, and if a prime ideal P contains an ideal A, then P is a factor of A. In the presence of the ascending chain condition, the following are also equivalent: (3) For every maximal ideal M , there is no ideal between M and M^2 ; (4) the lattice of ideals is distributive. These results generalize known conditions for an integral domain to be Dedekind. [Rings satisfying (1) have been studied by S. Mori, J. Sci. Hirosima Univ. Ser. A. 10, 117--136 (1940); these Rev. 2, 121.]
Reviewed by I. S. Cohen

2,121a 09.1X
Mori, Shinziro. Allgemeine Z.P.I.-Ringe.
J. Sci. Hirosima Univ. Ser. A. 10 (1940). 117--136.

A commutative ring R is termed a general Z.P.I. ring if every ideal in R can be expressed as a product of a finite number of prime ideals. Thus rings without unit element for multiplication and rings with divisors of zero are included in the class of rings considered by the author. As a main result the author proves that a ring R is a Z.P.I. ring if and only if (1) every ideal of R has a finite basis, (2) for every pair of maximal prime ideals P, P' (that is, R/P, R/P' are fields != 0) there is no ideal Q with PP' < Q < P, (3) there is no ideal Q with R^2 < Q < R. (The three conditions are independent.) This theorem essentially depends on the fact that in a Z.P.I. ring P P_1 = P if P < P_1 and if R/P is not a field. To prove the latter assertion it is necessary to investigate the relationship between the ideal theory of R and R/P. Finally the author formulates two theorems which are equivalent to his main theorem. For details and the methods of proof see the original paper.
Reviewed by O. F. G. Schilling

Zbl Google Translation of http://www.emis.de/cgi-bin/Zarchive?an=0024.00801 K. Kubo (s. this. Zbl. 23,102) characterized those commutative rings, in which every ideal from the whole ring and different from the zero-ideal ideal can be represented uniquely as product of finitely many prime ideals. The uniqueness idea is so sharply calm? that look for the occurrence of redundant (simply omitable), from the total ring different prime ideal factors to be excluded is. On this condition one (with more easily addition of the results won by Kubo themselves) receives the main clause: A commutative ring with unique prime ideal decomposition is either an integral domain, to which the well-known Noether five axioms apply, or a "primary, detachable ring", i.e. a ring with unit element, which contains only one prime ideal at ideals \p and its powers, whereby for a sufficiently large exponent \p^n = (0) becomes. -- By a Z.P.I. ring the author understands a commutative ring, which needs to be neither zero-divisor free nor contain unit element, and in which each ideal can be represented as product of finitely many prime ideals; Uniqueness of the representation is not demanded in contrast to the work by K. Kubo. The idea of the Z.P.I. ring is thus as far calm? as at all possible. As main results are emphasized: All Z.P.I. rings are O-rings, thus rings with maximum condition (divisor chain set). -- An O-ring with unit element is then and a Z.P.I. ring only if with no maximum ring prime ideal \p between \p and \p^2 a genuine intermediate ideal lies (the "Sono condition" characteristic of the Japanese direction of the abstract ideal theory). -- The Z.P.I. rings with unit element are nothing one but those already 1925 of W. Krull (S.-B. Heidelberg. Akad. Wiss. 1925, 5. Abhandl.) in their structure exactly described "multiplication rings with maximum condition". -- A Ring \R without unit element is then a Z.P.I. ring only if it possesses a direct decmomposition \R = \F + \m, whereby \F represents (possibly only from the nullelement existing) a field, while \m is an O-ring without unit element, which does not contain of (0) and \m different prime ideal, and in which between \m and \m^2 a genuine intermediate ideal does not lie.
Krull (Bonn).

-
Thank you, this answers my question although its means are beyond my math-skills :) – Listing Dec 14 '11 at 23:12
@Listing Gilmer's book is accessible to those who have mastered a university algebra course. You might find it enlightening to investigate how these general results specialize to your cases of interest. – Bill Dubuque Dec 14 '11 at 23:16