Prüfer ring

Prüfer ring

Definition.

A

commutative ring

R

with

non-zero unity

is a

Prüfer ring

(cf. Prüfer domain) if every

finitely generated

regular ideal

of

R

is

invertible

. (It can be proved that if every ideal of

R

generated by two elements is invertible, then all finitely generated ideals are invertible; cf.

invertibility of regularly generated ideal

.)

Denote generally by

𝔪

p

the

R

-module generated by the

coefficients

of a

polynomial

p

in

T

⁢

[

x

]

, where

T

is the

total ring of fractions

of

R

.  Such

coefficient modules

are, of course,

fractional ideals

of

R

.

Theorem 1 (Pahikkala 1982).

Let

R

be a commutative ring with non-zero unity and let

T

be the total ring of fractions of

R

.  Then,

R

is a Prüfer ring iff the

equation

𝔪

f

⁢

𝔪

g

=

𝔪

f

⁢

g

(1)

holds whenever

f

and

g

belong to the

polynomial ring

T

⁢

[

x

]

and at least one of the fractional ideals

𝔪

f

and

𝔪

g

is . (See also

product

of finitely generated ideals.)

Theorem 2 (Pahikkala 1982).

The commutative ring

R

with non-zero unity is Prüfer ring iff the

multiplication

rule

(

a

,

b

)

⁢

(

c

,

d

)

=

(

a

⁢

c

,

a

⁢

d

+

b

⁢

c

,

b

⁢

d

)

for the integral ideals of

R

holds whenever at least one of the

generators

a

,

b

,

c

and

d

is not

zero divisor

.

The proofs are found in the paper

J. Pahikkala 1982: “Some formulae for multiplying and inverting ideals”.  –

Annales universitatis turkuensis

183. Turun yliopisto (University of Turku).

Cf. the entries “

multiplication rule gives inverse ideal

(

http://planetmath.org/MultiplicationRuleGivesInverseIdeal

)” and “

two-generator property

(

http://planetmath.org/TwoGeneratorProperty

)”.

An additional characterization of Prüfer ring is found here in the entry “least common

multiple

(

http://planetmath.org/LeastCommonMultiple

)”, several other characterizations in [1] (p. 238–239).

Note.

A commutative ring

R

satisfying the equation (1) for all polynomials

f

,

g

is called a

Gaussian ring.

Thus any Prüfer domain (

http://planetmath.org/PruferDomain

) is always a Gaussian ring, and conversely (

http://planetmath.org/Converse

), an

integral domain

, which is a Gaussian ring, is a Prüfer domain.  Cf. [2].

References

1

M. Larsen & P. McCarthy:

Multiplicative

theory of ideals

.  Academic Press. New York (1971).

2

Sarah Glaz:

“The weak dimensions of Gaussian rings”. –

Proc. Amer. Math. Soc.

(2005).

Title

Prüfer ring

Canonical name

PruferRing

Date of creation

2015-05-05 15:21:07

Last modified on

2015-05-05 15:21:07

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

89

Author

pahio (2872)

Entry type

Theorem

Classification

msc 13C13

Classification

msc 13F05

Related topic

LeastCommonMultiple

Related topic

GeneratorsOfInverseIdeal

Related topic

ProductOfIdeals

Related topic

MultiplicationRing

Related topic

PruferDomain

Related topic

InvertibilityOfRegularlyGeneratedIdeal

Related topic

MultiplicationRuleGivesInverseIdeal

Related topic

ContentOfPolynomial

Related topic

ProductOfFinitelyGeneratedIdeals

Defines

Prüfer ring

Defines

coefficient module

Defines

Gaussian ring

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