product of finitely generated ideals

product of finitely generated ideals

Let

R

be a

commutative ring

having at least one

regular element

and

T

its

total ring of fractions

.  Let

𝔞

:=

(

a

0

,

a

1

,

…

,

a

m

-

1

)

and

𝔟

:=

(

b

0

,

b

1

,

…

,

b

n

-

1

)

be two

fractional ideals

of

R

(see the entry “fractional ideal of commutative ring”).  Then
the

product submodule

𝔞

⁢

𝔟

of

T

is also a
of

R

and is generated by all the elements

a

i

⁢

b

j

, thus having a

generating set

of

m

⁢

n

elements.

Such a generating set may be condensed in the case of any

Dedekind domain

, especially for the of any

algebraic number field

one has the

multiplication

formula

𝔞

⁢

𝔟

=

(

a

0

⁢

b

0

,

a

0

⁢

b

1

+

a

1

⁢

b

0

,

a

0

⁢

b

2

+

a

1

⁢

b

1

+

a

2

⁢

b

0

,

…

,

a

m

-

1

⁢

b

n

-

1

)

.

(1)

Here, the number of

generators

is only

m

+

n

-

1

(in principle, every ideal of a Dedekind domain has a generating system of two elements (

http://planetmath.org/TwoGeneratorProperty

)).  The formula is

characteristic

(

http://planetmath.org/Characterization

) still for a wider class of rings

R

which may contain

zero divisors

, viz. for the Prüfer rings (see [1]), but then at least one of

𝔞

and

𝔟

must be a

regular ideal

.

Note that the generators in (1) are formed similarly as the

coefficients

in the

product

of the

polynomials

f

⁢

(

X

)

:=

f

0

+

f

1

⁢

X

+

⋯

+

f

m

-

1

⁢

X

m

-

1

and

g

⁢

(

X

)

:=

g

0

+

g

1

⁢

X

+

⋯

+

g

n

-

1

⁢

X

n

-

1

.  Thus we may call the fractional ideals

𝔞

and

𝔟

of

R

the

coefficient modules

𝔪

f

and

𝔪

g

of the polynomials

f

and

g

(they are

R

-modules).  Hence the formula (1) may be rewritten as

𝔪

f

⁢

𝔪

g

=

𝔪

f

⁢

g

.

(2)

This formula says the same as Gauss’s lemma I for a

unique factorization domain

R

.

Arnold and Gilmer [2] have presented and proved the following generalisation of (2) which is valid under much less stringent assumptions than the ones requiring

R

to be a Prüfer ring (initially: a Prüfer domain); the proof is somewhat simplified in [1].

Theorem (Dedekind–Mertens lemma)

.
Let

R

be a

subring

of a commutative ring

T

.  If

f

and

g

are two arbitrary polynomials in the

polynomial ring

T

⁢

[

X

]

,
then there exists a non-negative

integer

n

such that the

R

-submodules of

T

generated by the coefficients of the
polynomials

f

,

g

and

f

⁢

g

satisfy the equality

𝔪

f

n

+

1

⁢

𝔪

g

=

𝔪

f

n

⁢

𝔪

f

⁢

g

.

(3)

References

1

J. Pahikkala:

“Some formulae for multiplying and inverting ideals”.  –

Ann. Univ. Turkuensis

183

(A) (1982).

2

J. Arnold & R. Gilmer:

“On the

contents of polynomials

”.  –

Proc. Amer. Math. Soc.

24

(1970).

Title

product of

finitely generated

ideals

Canonical name

ProductOfFinitelyGeneratedIdeals

Date of creation

2015-05-05 19:19:39

Last modified on

2015-05-05 19:19:39

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

29

Author

pahio (2872)

Entry type

Definition

Classification

msc 13A15

Classification

msc 16D25

Classification

msc 16D10

Synonym

special cases of ideal product

Related topic

PruferRing

Related topic

IdealGeneratorsInPruferRing

Related topic

IdealDecompositionInDedekindDomain

Related topic

EntriesOnFinitelyGeneratedIdeals

Related topic

UniqueFactorizationAndIdealsInRingOfIntegers

Related topic

ContentOfAPolynomial

Related topic

WellDefinednessOfProductOfFinitelyGeneratedIdeals

Defines

Dedekind–Mertens lemma

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