aarjaneiro/planetmath huggingface dataset:
quotient of ideals

quotient of ideals

Let

R

be a

commutative ring

having

regular elements

and let

T

be its

total ring of fractions

.  If

𝔞

and

𝔟

are

fractional ideals

of

R

, then one can define two different or

residuals

of

𝔞

by

𝔟

:

•

𝔞

:

𝔟

:=

{

r

∈

R

|

r

𝔟

⊆

𝔞

}

•

[

𝔞

:

𝔟

]

:=

{

t

∈

T

|

t

𝔟

⊆

𝔞

}

They both are fractional ideals of

R

, and the former in fact an integral ideal of

R

.  It is clear that

𝔞

:

𝔟

=

[

𝔞

:

𝔟

]

∩

R

.

In the special case that

R

has

non-zero unity

and

𝔟

has the inverse ideal

𝔟

-

1

, we have

[

𝔞

:

𝔟

]

=

𝔞

𝔟

-

1

,

in particular

[

R

:

𝔟

]

=

𝔟

-

1

.

Some rules concerning the former of

quotient

(the corresponding rules are valid also for the latter ):

1.

𝔞

⊆

𝔟

⇒

𝔞

:

𝔠

⊆

𝔟

:

𝔠

∧

𝔠

:

𝔞

⊇

𝔠

:

𝔟

2.

𝔞

:

(

𝔟

𝔠

)

=

(

𝔞

:

𝔟

)

:

𝔠

3.

𝔞

:

(

𝔟

+

𝔠

)

=

(

𝔞

:

𝔟

)

∩

(

𝔞

:

𝔠

)

4.

(

𝔞

∩

𝔟

)

:

𝔠

=

(

𝔞

:

𝔠

)

∩

(

𝔟

:

𝔠

)

Remark.

In a Prüfer ring

R

the addition (

http://planetmath.org/SumOfIdeals

) and

intersection

of ideals are dual operations of each other in the sense that there we have the duals

𝔞

:

(

𝔟

∩

𝔠

)

=

(

𝔞

:

𝔟

)

+

(

𝔞

:

𝔠

)

(

𝔞

+

𝔟

)

:

𝔠

=

(

𝔞

:

𝔠

)

+

(

𝔟

:

𝔠

)

of the two last rules if the are

finitely generated

.

Title

quotient of ideals

Canonical name

QuotientOfIdeals

Date of creation

2013-03-22 14:48:36

Last modified on

2013-03-22 14:48:36

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

20

Author

pahio (2872)

Entry type

Definition

Classification

msc 13B30

Synonym

residual

Synonym

quotient ideal

Related topic

SumOfIdeals

Related topic

ProductOfIdeals

Related topic

Submodule

Related topic

ArithmeticalRing

Generated on Fri Feb 9 17:27:54 2018 by

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rational number

rational number

The

rational numbers

ℚ

are the

fraction field

of the ring

ℤ

of

integers

. In more

elementary

terms, a rational number is a

quotient

a

/

b

of two integers

a

and

b

, where

b

is nonzero. Two

fractions

a

/

b

and

c

/

d

are

equivalent

if the

product

of the cross terms is equal:

a

b

=

c

d

⇔

a

⁢

d

=

b

⁢

c

Addition

and

multiplication

of fractions are given by the formulae

a

b

+

c

d

=

a

⁢

d

+

b

⁢

c

b

⁢

d

a

b

⋅

c

d

=

a

⁢

c

b

⁢

d

The

field of rational

numbers is an

ordered field

, under the

ordering relation

≤

defined as follows:

a

/

b

≤

c

/

d

if

1.

the

inequality

a

⋅

d

≤

b

⋅

c

holds in the integers, and

b

has the same sign as

d

, or

2.

the inequality

a

⋅

d

≥

b

⋅

c

holds in the integers, and

b

has the opposite sign as

d

.

Under this ordering relation, the rational numbers form a

topological space

under the

order topology

. The set of rational numbers is dense when considered as a subset of the

real numbers

.

Title

rational number

Canonical name

RationalNumber

Date of creation

2013-03-22 11:50:30

Last modified on

2013-03-22 11:50:30

Owner

djao (24)

Last modified by

djao (24)

Numerical id

15

Author

djao (24)

Entry type

Definition

Classification

msc 13B30

Classification

msc 11A99

Classification

msc 03E99

Synonym

ℚ

Related topic

Fraction

Related topic

ProofThatTheRationalsAreCountable

Defines

rational

Generated on Fri Feb 9 17:27:56 2018 by

LaTeXML
topic entry on rational numbers

topic entry on rational numbers

PlanetMath articles concerning the rational numbers

Definition

•

rational number

•

fraction

Properties

•

commensurable numbers

•

rational algebraic integers

Forms

•

partial fractions

•

any rational number is a sum of

unit fractions

•

Egyptian fractions

•

mixed fraction

•

decimal fraction

•

decimal expansion

•

order valuation

•

continued fraction

Where are rational numbers?

•

rational points on one dimensional sphere

•

rational sine and cosine

•

rational Briggsian logarithms of integers

•

irrational

to an irrational power can be rational

•

square roots of rationals

•

Roth’s

theorem

•

rational root theorem

•

canonical form of element of number field

•

http://

planetmath

.org/node/12127

solutions of the equation

x

y

=

y

x

Sets

•

rings of rational numbers

•

prime subfield

•

proof that the rationals are

countable

•

another proof of

cardinality

of the rationals

•

dense set

Polynomials and functions

•

Bernoulli polynomials and numbers

•

Euler polynomials

•

Dirichlet’s

function

•

a

pathological function of Riemann

Rational vs. irrational

•

theory of rational and irrational numbers

Title

topic entry on rational numbers

Canonical name

TopicEntryOnRationalNumbers

Date of creation

2013-03-22 19:08:15

Last modified on

2013-03-22 19:08:15

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

12

Author

pahio (2872)

Entry type

Topic

Classification

msc 13B30

Classification

msc 11A99

Classification

msc 03E99

Synonym

entries on rational numbers

Related topic

TopicEntryOnRealNumbers

Generated on Fri Feb 9 17:28:01 2018 by

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total ring of fractions

total ring of fractions

For a

commutative ring

R

having

regular elements

, we may form

T

=

S

-

1

⁢

R

, the

total ring of fractions

(

quotients

) of

R

, as the

localization

of

R

at

S

, where

S

is the set of all non-zero-divisors of

R

.  Then,

T

can be regarded as an

extension ring

of

R

(similarly as the

field of fractions

of an

integral domain

is an extension ring).

T

has the

non-zero unity

1.

Title

total ring of fractions

Canonical name

TotalRingOfFractions

Date of creation

2013-03-22 14:22:31

Last modified on

2013-03-22 14:22:31

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Definition

Classification

msc 13B30

Synonym

total ring of quotients

Related topic

ExtensionByLocalization

Related topic

FractionField

Generated on Fri Feb 9 17:28:02 2018 by

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I-adic topology

I

-adic topology

Let

R

be a ring and

I

an ideal in

R

such that

⋂

k

=

1

∞

I

k

=

{

0

}

.

Though not usually explicitly done, we can define a metric on

R

by defining

o

⁢

r

⁢

d

I

⁢

(

r

)

for a

r

∈

R

by

o

⁢

r

⁢

d

I

⁢

(

r

)

=

k

where

k

is the largest integer such that

r

∈

I

k

(well-defined by the

intersection

assumption

, and

I

0

is taken to be the entire ring) and by

o

⁢

r

⁢

d

I

⁢

(

0

)

=

∞

, and then defining for any

r

1

,

r

2

∈

R

,

d

I

⁢

(

r

1

,

r

2

)

=

2

-

o

⁢

r

⁢

d

I

⁢

(

r

1

-

r

s

)

.

The

topology

induced

by this metric is called the

I

-adic topology. Note that the number 2 was chosen rather arbitrarily. Any other

real number

greater than 1 will induce an

equivalent

topology.

Except in the case of the similarly-defined

p

-adic topology, it is rare that reference is made to the actual

I

-adic metric. Instead, we usually refer to the

I

-adic topology.

In particular, a

sequence

of

elements

in

{

r

i

}

∈

R

is Cauchy with respect to this topology if for any

k

there exists an

N

such that for all

m

,

n

≥

N

we have

(

a

m

-

a

n

)

∈

I

k

. (Note the parallel with the metric version of Cauchy, where

k

plays the part analogous to an arbitrary

ϵ

). The ring

R

is

complete

with respect to the

I

-adic topology if every such

Cauchy sequence

converges

to an

element of

R

.

Title

I

-adic topology

Canonical name

IadicTopology

Date of creation

2013-03-22 14:36:59

Last modified on

2013-03-22 14:36:59

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 13B35

Synonym

I-adic topology

Generated on Fri Feb 9 17:28:04 2018 by

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algebra (module)

algebra (module)

Given a

commutative ring

R

, an algebra over

R

is a
module

M

over

R

, endowed with a law of composition

f

:

M

×

M

→

M

which is

R

-

bilinear

.

Most of the important

algebras

in mathematics belong to
one or the other of two classes: the unital

associative
algebras

, and the

Lie algebras

.

1

Unital associative algebras

In these cases, the “product” (as it is called) of two elements

v

and

w

of the module, is denoted simply by

v

⁢

w

or

v

∙

w

or the
like.

Any unital associative algebra is an algebra in the sense of djao (a
sense which is also used by Lang in his book

Algebra

(Springer-Verlag)).

Examples of unital associative algebras:

–

tensor algebras

and quotients of them

–

Cayley algebras

, such as the ring of

quaternions

–

polynomial rings

– the

ring of endomorphisms

of a

vector space

, in which the bilinear
product of two mappings is simply the

composite

mapping.

2

Lie algebras

In these cases the bilinear product is denoted by

[

v

,

w

]

,
and satisfies

[

v

,

v

]

=

0

⁢

for all

⁢

v

∈

M

[

v

,

[

w

,

x

]

]

+

[

w

,

[

x

,

v

]

]

+

[

x

,

[

v

,

w

]

]

=

0

⁢

for all

⁢

v

,

w

,

x

∈

M

The second of these formulas is called the

Jacobi identity

. One proves
easily

[

v

,

w

]

+

[

w

,

v

]

=

0

⁢

for all

⁢

v

,

w

∈

M

for any Lie algebra M.

Lie algebras arise naturally from Lie groups, q.v.

Title

algebra (module)

Canonical name

Algebramodule

Date of creation

2013-03-22 13:20:50

Last modified on

2013-03-22 13:20:50

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Definition

Classification

msc 13B99

Classification

msc 20C99

Classification

msc 16S99

Defines

Jacobi identity

Generated on Fri Feb 9 17:28:06 2018 by

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basic algebra

basic algebra

Let

A

be a finite dimensional, unital algebra over a field

k

. By

Krull-Schmidt Theorem

A

can be decomposed as a (right)

A

-module as follows:

A

≃

P

1

⊕

⋯

⊕

P

k

where each

P

i

is an

indecomposable module

and this decomposition is unique.

Definition.

The

algebra

A

is called

(right) basic

if

P

i

is not

isomorphic

to

P

j

when

i

≠

j

.

Of course we may easily define what does it mean for algebra to be left basic. Fortunetly these properties coincide. Let as state some known facts (originally can be found in

[

1

]

):

Proposition

.

1.

A

finite algebra

A

over a field

k

is basic if and only if the algebra

A

/

rad

⁢

A

is isomorphic to a

product

of fields

k

×

⋯

×

k

. Thus

A

is right basic iff it is left basic;

2.

Every

simple module

over a

basic algebra

is one-dimensional;

3.

For any finite-dimensional, unital algebra

A

over

k

there exists finite-dimensional, unital, basic algebra

B

over

k

such that the

category

of finite-dimensional modules over

A

is

k

-linear

equivalent

to the category of finite-dimensional modules over

B

;

4.

Let

A

be a finite-dimensional, basic and connected (i.e. cannot be written as a product of nontrivial algebras) algebra over a field

k

. Then there exists a bound quiver

(

Q

,

I

)

such that

A

≃

k

⁢

Q

/

I

;

5.

If

(

Q

,

I

)

is a bound quiver over a field

k

, then both

k

⁢

Q

and

k

⁢

Q

/

I

are basic algebras.

References

1

I. Assem, D. Simson, A. Skowronski,

Elements of the

Representation Theory

of

Associative Algebras

, vol 1.

, Cambridge University Press 2006, 2007

Title

basic algebra

Canonical name

BasicAlgebra

Date of creation

2013-03-22 19:17:10

Last modified on

2013-03-22 19:17:10

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

5

Author

joking (16130)

Entry type

Definition

Classification

msc 13B99

Classification

msc 20C99

Classification

msc 16S99

Generated on Fri Feb 9 17:28:08 2018 by

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every ring is an integer algebra

every ring is an integer algebra

Let

R

be a ring. Then

R

is also an algebra over the

ring of integers

if we define the action of

ℤ

on

R

by the following rules:

0

⋅

x

=

0

(

n

+

1

)

⋅

x

=

n

⋅

x

+

x

(

-

n

)

⋅

x

=

-

(

n

⋅

x

)

In other words, the action of a

positive integer

n

on

x

is to add

x

to itself

n

times and the action of a negative integer

n

on

x

is to subtract

x

to itself

n

times.

Title

every ring is an

integer

algebra

Canonical name

EveryRingIsAnIntegerAlgebra

Date of creation

2013-03-22 14:47:47

Last modified on

2013-03-22 14:47:47

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Example

Classification

msc 13B99

Classification

msc 20C99

Classification

msc 16S99

Related topic

GeneralAssociativity

Generated on Fri Feb 9 17:28:10 2018 by

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modules over algebars and homomorphisms between them

modules over algebars and homomorphisms between them

Let

R

be a ring and let

A

be an

associative algebra

(not necessarily unital).

Definition.

A (left)

A

-module over

R

is a pair

(

M

,

∘

)

where

M

is a (left)

R

-module and

∘

:

A

×

M

→

M

is a

R

-

bilinear map

such that the following conditions hold:

1.

(

a

∘

b

)

∘

x

=

a

∘

(

b

∘

x

)

2.

r

⁢

(

a

∘

x

)

=

(

r

⁢

a

)

∘

x

=

a

∘

(

r

⁢

x

)

for any

a

,

b

∈

A

,

x

∈

M

and

r

∈

R

. We will simply use capital letters to denote modules.

Let

M

be an

A

-module over

R

. If

M

′

⊆

M

and

A

′

⊆

A

then by

A

′

⁢

M

′

we denote

R

-

submodule

of

M

generated by elements of the form

a

⁢

m

for

a

∈

A

′

and

m

∈

M

′

. We will call

M

unitary

if

A

⁢

M

=

M

. Note, that if

A

has

multiplicative identity

1

, then

M

is unitary if and only if

1

⁢

m

=

m

for any

m

∈

M

.

The reason we use name ,,

A

-module over

R

” instead of ,,

A

-module” is that these to

concepts

may differ. The latter means that we treat

A

simply as a ring and take modules over it. But such module need not be equiped with a ,,good”

R

-module

structure

. On the other hand this is always the case, when

M

is unitary over unital

algebra

.

If

M

and

N

are two

A

-modules over

R

, then a

function

f

:

M

→

N

is called an

A

-

homomorphism

iff

f

is an

R

-homomorphism and additionaly

f

⁢

(

a

⁢

m

)

=

a

⁢

f

⁢

(

m

)

for any

a

∈

A

and

m

∈

M

.

It can be easily checked that

A

-modules over

R

together with

A

-homomorphisms form a

category

which is

abelian

. Furthermore, if

A

is unital, then its

full subcategory

consisting unitary

R

-modules over

A

is

equivalent

to category of unitary

A

-modules.

In most cases it is important to assume that the base ring

R

is a field, even

algebraically closed

.

Title

modules over algebars and homomorphisms between them

Canonical name

ModulesOverAlgebarsAndHomomorphismsBetweenThem

Date of creation

2013-03-22 19:16:32

Last modified on

2013-03-22 19:16:32

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

5

Author

joking (16130)

Entry type

Definition

Classification

msc 13B99

Classification

msc 20C99

Classification

msc 16S99

Generated on Fri Feb 9 17:28:12 2018 by

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example of free module

example of free module

from the

definition

,

ℤ

n

is

http://planetmath.org/node/FreeModule

free as a

ℤ

-module for any positive integer

n

.

A more interesting example is the following:

Theorem 1

.

The set of rational numbers

Q

do

not

form a

http://planetmath.org/node/FreeModule

free

Z

-module.

Proof.

First note that any two

elements

in

ℚ

are

ℤ

-

linearly dependent

. If

x

=

p

1

q

1

and

y

=

p

2

q

2

, then

q

1

⁢

p

2

⁢

x

-

q

2

⁢

p

1

⁢

y

=

0

. Since basis (

http://planetmath.org/Basis

) elements
must be

linearly independent

, this shows that any basis must consist
of only one element, say

p

q

, with

p

and

q

relatively prime, and without loss of generality,

q

>

0

. The

ℤ

-span of

{

p

q

}

is the
set of rational numbers of the form

n

⁢

p

q

. I claim that

1

q

+

1

is not in the set. If it were, then we would have

n

⁢

p

q

=

1

q

+

1

for some

n

, but this implies that

n

⁢

p

=

q

q

+

1

which has no solutions for

n

,

p

∈

ℤ

,

q

∈

ℤ

+

, giving us
a

contradiction

.
∎

Title

example of free module

Canonical

name

ExampleOfFreeModule

Date of creation

2013-03-22 13:48:41

Last modified on

2013-03-22 13:48:41

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Example

Classification

msc 13C10

Generated on Fri Feb 9 17:28:31 2018 by

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torsion element

torsion element

Let

R

be a

commutative ring

, and

M

an

R

-module. We call an element

m

∈

M

a

torsion element

if there exists a non-zero-divisor

α

∈

R

such that

α

⋅

m

=

0

. The set is denoted by

t

⁢

o

⁢

r

⁢

(

M

)

.

t

⁢

o

⁢

r

⁢

(

M

)

is not empty since

0

∈

t

⁢

o

⁢

r

⁢

(

M

)

. Let

m

,

n

∈

t

⁢

o

⁢

r

⁢

(

M

)

, so there exist

α

,

β

≠

0

∈

R

such that

0

=

α

⋅

m

=

β

⋅

n

. Since

α

⁢

β

⋅

(

m

-

n

)

=

β

⋅

α

⋅

m

-

α

⋅

β

⋅

n

=

0

,

α

⁢

β

≠

0

, this implies that

m

-

n

∈

t

⁢

o

⁢

r

⁢

(

M

)

. So

t

⁢

o

⁢

r

⁢

(

M

)

is a

subgroup

of

M

. Clearly

τ

⋅

m

∈

t

⁢

o

⁢

r

⁢

(

M

)

for any non-zero

τ

∈

R

. This shows that

t

⁢

o

⁢

r

⁢

(

M

)

is a

submodule

of

M

, the

torsion submodule

of

M

. In particular, a module that equals its own torsion submodule is said to be a

torsion module

.

Title

torsion element

Canonical name

TorsionElement

Date of creation

2013-03-22 13:54:41

Last modified on

2013-03-22 13:54:41

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 13C12

Defines

torsion submodule

Defines

torsion module

Generated on Fri Feb 9 17:28:33 2018 by

LaTeXML
ideal generators in Prüfer ring

ideal generators in Prüfer ring

Let

R

be a Prüfer ring with

total ring of fractions

T

.
Let

𝔞

and

𝔟

be

fractional ideals

of

R

, generated by (

http://planetmath.org/IdealGeneratedByASet

)

m

and

n

elements of

T

, respectively.

•

Then the

sum ideal

𝔞

+

𝔟

may, of course, be generated by

m

+

n

elements.

•

If

𝔞

or

𝔟

is

regular

(

http://planetmath.org/FractionalIdealOfCommutativeRing

), then the

product

(

http://planetmath.org/ProductOfIdeals

) ideal

𝔞

⁢

𝔟

may be generated by

m

+

n

-

1

elements, since in Prüfer rings the

(

a

1

,

…

,

a

m

)

⁢

(

b

1

,

…

,

b

n

)

=

(

a

1

⁢

b

1

,

a

1

⁢

b

2

+

a

2

⁢

b

1

,

a

1

⁢

b

3

+

a

2

⁢

b

2

+

a

3

⁢

b

1

,

…

,

a

m

⁢

b

n

)

holds.

•

If both

𝔞

and

𝔟

are

regular ideals

, then the

intersection

𝔞

∩

𝔟

and the

quotient ideal

𝔞

:

𝔟

=

{

r

∈

R

|

r

𝔟

⊆

𝔞

}

both may be generated by

m

+

n

elements.

•

If

𝔞

is regular,  then it is also invertible (

http://planetmath.org/InvertibleIdeal

).  Its ideal has the expression (

http://planetmath.org/QuotientOfIdeals

)

𝔞

-

1

=

[

R

:

𝔞

]

=

{

t

∈

T

|

t

𝔞

⊆

R

}

and may be generated by

m

elements of

T

(see the

generators of inverse ideal

).

Cf. also the

two-generator property

.

References

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

-

Annales universitatis turkuensis

183.  Turun yliopisto (University of Turku) 1982.

Title

ideal

generators

in Prüfer ring

Canonical name

IdealGeneratorsInPruferRing

Date of creation

2013-03-22 14:33:04

Last modified on

2013-03-22 14:33:04

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

20

Author

pahio (2872)

Entry type

Result

Classification

msc 13C13

Related topic

FractionalIdeal

Related topic

ProductOfFinitelyGeneratedIdeals

Generated on Fri Feb 9 17:28:35 2018 by

LaTeXML
ideal inverting in Prüfer ring

ideal inverting in Prüfer ring

Theorem

.

Let

𝔞

1

, …,

𝔞

n

be

invertible

fractional ideals

of a Prüfer ring.  Then also their sum and

intersection

are invertible, and the inverse ideals of these are obtained by the formulae resembling de Morgan’s laws:

(

𝔞

1

+

⋯

+

𝔞

n

)

-

1

=

𝔞

1

-

1

∩

⋯

∩

𝔞

n

-

1

(

𝔞

1

∩

⋯

∩

𝔞

n

)

-

1

=

𝔞

1

-

1

+

⋯

+

𝔞

n

-

1

This is due to the fact, that the sum of any ideals is the smallest ideal containing these ideals and the intersection of the ideals is the largest ideal

contained

in each of these ideals.  Cf.

sum of ideals

,

quotient of ideals

.

References

1

J. Pahikkala

:  “Some formulae for multiplying and inverting ideals”.

-

Annales universitatis turkuensis

183

.  Turun yliopisto (University of Turku) 1982.

Title

ideal inverting in Prüfer ring

Canonical name

IdealInvertingInPruferRing

Date of creation

2015-05-06 14:34:48

Last modified on

2015-05-06 14:34:48

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

15

Author

pahio (2872)

Entry type

Theorem

Classification

msc 13C13

Related topic

DualityInMathematics

Related topic

DualityOfGudermannianAndItsInverseFunction

Generated on Fri Feb 9 17:28:37 2018 by

LaTeXML
Cohen-Macaulay module

Cohen-Macaulay module

A module

M

over a ring

R

is a

Cohen-Macaulay module

if its depth is defined and equals its

Krull dimension

. A ring is said to be Cohen-Macaulay (or just C-M) if it is a Cohen-Macaulay module viewed as a module over itself.

Cohen-Macaulay rings

are used extensively in combinatorial geometry and

commutative ring

theory, and has applications to

algebraic geometry

as well. For instance, a

variety

all of whose

local rings

are Cohen-Macaulay has, in a sense, nicer behaviour than an arbitrary singular variety.

Title

Cohen-Macaulay module

Canonical

name

CohenMacaulayModule

Date of creation

2013-03-22 14:14:58

Last modified on

2013-03-22 14:14:58

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Definition

Classification

msc 13C14

Classification

msc 16E65

Defines

Cohen-Macaulay ring

Defines

C-M module

Defines

C-M ring

Generated on Fri Feb 9 17:28:41 2018 by

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Ulrich module

Ulrich module

A

maximal

Cohen-Macaulay module

M

over a

Noetherian

local ring

(

R

,

𝔪

,

k

)

is Ulrich if

e

⁢

(

M

)

=

μ

⁢

(

M

)

, where

e

⁢

(

M

)

is the Hilbert-Samuel multiplicity of

M

and

μ

⁢

(

M

)

is the minimal number of

generators

of

M

. When

M

is a maximal Cohen-Macaulay module and

𝔪

has a minimal

reduction

I

generated by a system of parameters,

M

is Ulrich if and only if

𝔪

⁢

M

=

I

⁢

M

.

Title

Ulrich module

Canonical name

UlrichModule

Date of creation

2013-03-22 18:13:38

Last modified on

2013-03-22 18:13:38

Owner

yshen (21076)

Last modified by

yshen (21076)

Numerical id

8

Author

yshen (21076)

Entry type

Definition

Classification

msc 13C14

Generated on Fri Feb 9 17:28:42 2018 by

LaTeXML
bound on the Krull dimension of polynomial rings

bound on the Krull dimension of polynomial rings

If

A

is a

commutative ring

, and

dim

denotes

Krull dimension

, then

dim

⁡

(

A

)

+

1

≤

dim

⁡

(

A

⁢

[

x

]

)

≤

2

⁢

dim

⁡

(

A

)

+

1

.

It is known (see

[

Seid

]

,

[

Seid2

]

) that for any

k

≥

0

and

n

with

k

+

1

≤

n

≤

2

⁢

k

+

1

, there exists a ring

A

such that

dim

⁡

A

=

k

and

dim

⁡

A

⁢

[

x

]

=

n

.

References

Seid

A. Seidenberg,

A note on the dimension theory of rings.

Pacific J. of Mathematics, Volume 3 (1953), 505-512.

Seid2

A. Seidenberg,

On the dimension theory of rings (II).

Pacific J. of Mathematics, Volume 4 (1954), 603-614.

Title

bound on the Krull dimension of

polynomial rings

Canonical name

BoundOnTheKrullDimensionOfPolynomialRings

Date of creation

2013-03-22 15:22:11

Last modified on

2013-03-22 15:22:11

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 13C15

Generated on Fri Feb 9 17:28:44 2018 by

LaTeXML