Intereting Posts

if locally Lipschitz implies Lipschitz on compacts.
(Fast way to) Get a combination given its position in (reverse-)lexicographic order
Direct image of vector bundle
Prove that $ \lim_{n \to \infty}\frac{n}{\sqrt n!}=e$?
Equation of a sphere as the determinant of its variables and sampled points
Roots of polynomials on the unit circle
Finding $P$ such that $P^TAP$ is a diagonal matrix
Another trigonometric proof…?
Radius, diameter and center of graph
Help with $\int\frac{1}{1+x^8}dx$
volume of a truncated cone that is not a frustum
Why is the following example of a Markov process not strong Markov
How to compute the determinant of a tridiagonal matrix with constant diagonals?
All natural numbers are equal.
If multiplication is not repeated addition

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?

- Cogroup structures on the profinite completion of the integers
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- Infinite Groups with Finitely many Conjugacy Classes
- Isomorphism $k/(y-x^2)$ onto $k$
- Is the identity matrix the only matrix which is its own inverse?

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.

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- Euler's remarkable prime-producing polynomial and quadratic UFDs
- Prove every element of $G$ has finite order.
- The Maximum possible order for an element $S_n$
- A Laskerian non-Noetherian ring
- Image of ring homomorphism $\phi : \mathbb{Z} \to \mathbb{Q}$?
- If $x^3 =x$ then $6x=0$ in a ring

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).

See also the following MR’s (not meant to be exhaustive).

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).

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