aarjaneiro/planetmath huggingface dataset:
topological ring

topological ring

A ring

R

which is a

topological space

is called a

topological ring

if the

addition

,

multiplication

, and the

additive inverse

functions are

continuous functions

from

R

×

R

to

R

.

A

topological division ring

is a topological ring such that the

multiplicative inverse

function is

continuous

away from

0

. A

topological field

is a topological division ring that is a field.

Remark

. It is easy to see that if

R

contains the multiplicative identity

1

, then

R

is a topological ring iff addition and multiplication are continuous. This is true because the additive inverse of an element can be written as the

product

of the element and

-

1

. However, if

R

does not contain

1

, it is necessary to impose the

continuity

condition on the additive inverse operation.

Title

topological ring

Canonical name

TopologicalRing

Date of creation

2013-03-22 12:45:59

Last modified on

2013-03-22 12:45:59

Owner

djao (24)

Last modified by

djao (24)

Numerical id

6

Author

djao (24)

Entry type

Definition

Classification

msc 12J99

Classification

msc 13J99

Classification

msc 54H13

Related topic

TopologicalGroup

Related topic

TopologicalVectorSpace

Defines

topological field

Defines

topological division ring

Generated on Fri Feb 9 16:45:16 2018 by

LaTeXML
proof of Vieta’s formula

proof of Vieta’s formula

Proof:

We can write

P

⁢

(

x

)

as

P

⁢

(

x

)

=

a

n

⁢

(

x

-

r

1

)

⁢

(

x

-

r

2

)

⁢

…

⁢

(

x

-

r

n

)

=

a

n

⁢

(

x

n

-

S

1

⁢

x

n

-

1

+

S

2

⁢

x

n

-

2

-

…

+

(

-

1

)

n

⁢

S

n

)

.

Since we also have that

P

⁢

(

x

)

=

a

n

⁢

x

n

+

a

n

-

1

⁢

x

n

-

1

+

…

⁢

a

1

⁢

x

+

a

0

,

setting the coefficients equal yields

a

n

⁢

(

-

1

)

i

⁢

S

i

=

a

n

-

i

which is what the theorem stated.

Title

proof of Vieta’s formula

Canonical name

ProofOfVietasFormula

Date of creation

2013-03-22 15:26:59

Last modified on

2013-03-22 15:26:59

Owner

neapol1s (9480)

Last modified by

neapol1s (9480)

Numerical id

4

Author

neapol1s (9480)

Entry type

Proof

Classification

msc 12Y05

Generated on Fri Feb 9 16:45:18 2018 by

LaTeXML
Vieta’s formula

Vieta’s formula

Suppose

P

⁢

(

x

)

is a

polynomial

of degree

n

with roots

r

1

,

r

2

,

…

,

r

n

(not necessarily distinct). For

1

≤

k

≤

n

, define

S

k

by

S

k

=

∑

1

≤

α

1

<

α

2

<

…

⁢

α

k

≤

n

r

α

1

⁢

r

α

2

⁢

…

⁢

r

α

k

For example,

S

1

=

r

1

+

r

2

+

r

3

+

…

+

r

n

S

2

=

r

1

⁢

r

2

+

r

1

⁢

r

3

+

r

1

⁢

r

4

+

r

2

⁢

r

3

+

…

+

r

n

-

1

⁢

r

n

Then writing

P

⁢

(

x

)

as

P

⁢

(

x

)

=

a

n

⁢

x

n

+

a

n

-

1

⁢

x

n

-

1

+

…

⁢

a

1

⁢

x

+

a

0

,

we find that

S

i

=

(

-

1

)

i

⁢

a

n

-

i

a

n

For example, if

P

⁢

(

x

)

is a polynomial of degree 1, then

P

⁢

(

x

)

=

a

1

⁢

x

+

a

0

and clearly

r

1

=

-

a

0

a

1

.

If

P

⁢

(

x

)

is a polynomial of degree 2, then

P

⁢

(

x

)

=

a

2

⁢

x

2

+

a

1

⁢

x

+

a

0

and

r

1

+

r

2

=

-

a

1

a

2

and

r

1

⁢

r

2

=

a

0

a

2

. Notice that both of these formulas can be determined from the

quadratic formula

.

More intrestingly, if

P

⁢

(

x

)

=

a

3

⁢

x

3

+

a

2

⁢

x

2

+

a

1

⁢

x

+

a

0

, then

r

1

+

r

2

+

r

3

=

-

a

2

a

3

,

r

1

⁢

r

2

+

r

2

⁢

r

3

+

r

3

⁢

r

1

=

a

1

a

3

, and

r

1

⁢

r

2

⁢

r

3

=

-

a

0

a

3

.

Title

Vieta’s formula

Canonical name

VietasFormula

Date of creation

2013-03-22 15:21:55

Last modified on

2013-03-22 15:21:55

Owner

neapol1s (9480)

Last modified by

neapol1s (9480)

Numerical id

9

Author

neapol1s (9480)

Entry type

Theorem

Classification

msc 12Y05

Related topic

PropertiesOfQuadraticEquation

Generated on Fri Feb 9 16:45:20 2018 by

LaTeXML
absolute value

absolute value

Let

R

be an

ordered ring

and let

a

∈

R

. The

absolute value

of

a

is defined to be the

function

|

|

:

R

→

R

given by

|

a

|

:=

{

a

if

⁢

a

≥

0

,

-

a

otherwise.

In particular, the usual absolute value

|

|

on the field

ℝ

of

real numbers

is defined in this manner. An

equivalent

definition over the real numbers is

|

a

|

:=

max

⁡

{

a

,

-

a

}

.

Absolute value has a different meaning in the case of

complex numbers

: for a complex number

z

∈

ℂ

, the absolute value

|

z

|

of

z

is defined to be

x

2

+

y

2

, where

z

=

x

+

y

⁢

i

and

x

,

y

∈

ℝ

are real.

All absolute value functions satisfy the defining properties of a

valuation

, including:

•

|

a

|

≥

0

for all

a

∈

R

, with equality if and only if

a

=

0

•

|

a

⁢

b

|

=

|

a

|

⋅

|

b

|

for all

a

,

b

∈

R

•

|

a

+

b

|

≤

|

a

|

+

|

b

|

for all

a

,

b

∈

R

(

triangle inequality

)

However, in general they are not literally valuations, because valuations are required to be real valued. In the case of

ℝ

and

ℂ

, the absolute value is a valuation, and it induces a metric in the usual way, with distance function defined by

d

⁢

(

x

,

y

)

:=

|

x

-

y

|

.

Title

absolute value

Canonical name

AbsoluteValue

Date of creation

2013-03-22 11:52:09

Last modified on

2013-03-22 11:52:09

Owner

djao (24)

Last modified by

djao (24)

Numerical id

10

Author

djao (24)

Entry type

Definition

Classification

msc 13-00

Classification

msc 11A15

Generated on Fri Feb 9 17:18:53 2018 by

LaTeXML
associates

associates

Two

elements

in a

ring with unity

are

associates

or

associated elements

of each other
if one can be obtained from the other by multiplying by some unit,
that is,

a

and

b

are associates if there is a unit

u

such that

a

=

b

⁢

u

.  Equivalently, one can say that two associates are divisible by each other.

The

binary relation

“is an associate of” is an

equivalence relation

on any ring with unity. For example, the

equivalence class

of the
unity of the ring consists of all units of the ring.

Examples.

In the ring

ℤ

of the rational integers, only opposite numbers

±

n

are associates.  Among the

polynomials

, the associates of a polynomial are gotten by multiplying the polynomial by an element belonging to the

coefficient ring

in question (and being no

zero divisor

).

In an

integral domain

, two elements are associates if and only if they

generate

the same

principal ideal

.

Title

associates

Canonical name

Associates

Date of creation

2013-03-22 11:56:31

Last modified on

2013-03-22 11:56:31

Owner

drini (3)

Last modified by

drini (3)

Numerical id

10

Author

drini (3)

Entry type

Definition

Classification

msc 13-00

Classification

msc 16-00

Related topic

Ring

Related topic

Unit

Defines

associate

Defines

associated element

Generated on Fri Feb 9 17:18:56 2018 by

LaTeXML
cancellation ring

cancellation ring

A ring

R

is a

cancellation ring

if for all

a

,

b

∈

R

, if

a

⋅

b

=

0

then either

a

=

0

or

b

=

0

.

Title

cancellation ring

Canonical

name

CancellationRing

Date of creation

2013-03-22 11:51:22

Last modified on

2013-03-22 11:51:22

Owner

djao (24)

Last modified by

djao (24)

Numerical id

8

Author

djao (24)

Entry type

Definition

Classification

msc 13-00

Related topic

IntegralDomain

Related topic

ZeroDivisor

Generated on Fri Feb 9 17:18:59 2018 by

LaTeXML
every prime ideal is radical

every prime ideal is radical

Let

ℛ

be a

commutative ring

and let

𝔓

be
a

prime ideal

of

ℛ

.

Proposition 1

.

Every prime ideal

P

of

R

is a

radical
ideal

, i.e.

𝔓

=

Rad

⁢

(

𝔓

)

Proof.

Recall that

𝔓

⊊

ℛ

is a prime ideal
if and only if for any

a

,

b

∈

ℛ

a

⋅

b

∈

𝔓

⇒

a

∈

𝔓

⁢

or

⁢

b

∈

𝔓

Also, recall that

Rad

⁡

(

𝔓

)

=

{

r

∈

ℛ

∣

∃

n

∈

ℕ

⁢

such that

⁢

r

n

∈

𝔓

}

Obviously, we have

𝔓

⊆

Rad

⁡

(

𝔓

)

(just take

n

=

1

), so it remains
to show the reverse

inclusion

.

Suppose

r

∈

Rad

⁡

(

𝔓

)

, so there exists
some

n

∈

ℕ

such that

r

n

∈

𝔓

. We want to
prove that

r

must be an element of the prime ideal

𝔓

. For this, we use

induction

on

n

to prove the
following

proposition

:

For all

n

∈

ℕ

, for all

r

∈

ℛ

,

r

n

∈

𝔓

⇒

r

∈

𝔓

.

Case

n

=

1

:

This is clear,

r

∈

𝔓

⇒

r

∈

𝔓

.

Case

n

⇒

Case

n

+

1

:

Suppose we have proved
the proposition for the case

n

, so our

induction hypothesis

is

∀

r

∈

ℛ

,

r

n

∈

𝔓

⇒

r

∈

𝔓

and suppose

r

n

+

1

∈

𝔓

. Then

r

⋅

r

n

=

r

n

+

1

∈

𝔓

and since

𝔓

is a prime ideal we have

r

∈

𝔓

⁢

or

⁢

r

n

∈

𝔓

Thus we conclude, either directly or using the induction
hypothesis, that

r

∈

𝔓

as desired.

∎

Title

every

prime ideal is radical

Canonical

name

EveryPrimeIdealIsRadical

Date of creation

2013-03-22 13:56:54

Last modified on

2013-03-22 13:56:54

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Theorem

Classification

msc 13-00

Classification

msc 14A05

Synonym

prime ideal is radical

Related topic

RadicalOfAnIdeal

Related topic

PrimeIdeal

Related topic

HilbertsNullstellensatz

Generated on Fri Feb 9 17:19:03 2018 by

LaTeXML
module

module

Let

R

be a ring with identity. A

left module

M

over

R

is a set with two binary operations,

+

:

M

×

M

⟶

M

and

⋅

:

R

×

M

⟶

M

, such that

1.

(

𝐮

+

𝐯

)

+

𝐰

=

𝐮

+

(

𝐯

+

𝐰

)

for all

𝐮

,

𝐯

,

𝐰

∈

M

2.

𝐮

+

𝐯

=

𝐯

+

𝐮

for all

𝐮

,

𝐯

∈

M

3.

There exists an element

𝟎

∈

M

such that

𝐮

+

𝟎

=

𝐮

for all

𝐮

∈

M

4.

For any

𝐮

∈

M

, there exists an element

𝐯

∈

M

such that

𝐮

+

𝐯

=

𝟎

5.

a

⋅

(

b

⋅

𝐮

)

=

(

a

⋅

b

)

⋅

𝐮

for all

a

,

b

∈

R

and

𝐮

∈

M

6.

a

⋅

(

𝐮

+

𝐯

)

=

(

a

⋅

𝐮

)

+

(

a

⋅

𝐯

)

for all

a

∈

R

and

𝐮

,

𝐯

∈

M

7.

(

a

+

b

)

⋅

𝐮

=

(

a

⋅

𝐮

)

+

(

b

⋅

𝐮

)

for all

a

,

b

∈

R

and

𝐮

∈

M

A left module

M

over

R

is called

unitary

or

unital

if

1

R

⋅

𝐮

=

𝐮

for all

𝐮

∈

M

.

A (unitary or unital)

right module

is defined analogously, except that the

function

⋅

goes from

M

×

R

to

M

and the

scalar multiplication

operations act on the right. If

R

is

commutative

, there is an equivalence of categories between the category of left

R

–modules and the category of right

R

–modules.

Title

module

Canonical name

Module

Date of creation

2013-03-22 11:49:14

Last modified on

2013-03-22 11:49:14

Owner

djao (24)

Last modified by

djao (24)

Numerical id

11

Author

djao (24)

Entry type

Definition

Classification

msc 13-00

Classification

msc 16-00

Classification

msc 20-00

Classification

msc 44A20

Classification

msc 33E20

Classification

msc 30D15

Synonym

left module

Synonym

right module

Related topic

MaximalIdeal

Related topic

VectorSpace

Generated on Fri Feb 9 17:19:07 2018 by

LaTeXML
multiplicative order of an integer modulo m

multiplicative order of an integer modulo m

Definition

.

Let

m

>

1

be an

integer

and let

a

be another integer relatively prime to

m

. The order (

http://planetmath.org/OrderGroup

) of

a

modulo

m

(or the

multiplicative order

of

a

mod

m

) is the smallest

positive integer

n

such that

a

n

≡

1

mod

m

. The order is sometimes denoted by

ord

⁡

a

or

ord

m

⁡

a

.

Remarks

.

Several remarks are in order:

1.

Notice that if

gcd

⁡

(

a

,

m

)

=

1

then

a

belong to the units

(

ℤ

/

m

⁢

ℤ

)

×

of

ℤ

/

m

⁢

ℤ

. The units

(

ℤ

/

m

⁢

ℤ

)

×

form a

group with

respect to

multiplication

, and the number of

elements

in the

subgroup

generated by

a

(and its powers) is the order of the integer

a

modulo

m

.

2.

By Euler’s

theorem

,

a

ϕ

⁢

(

m

)

≡

1

mod

m

, therefore the order of

a

is less or equal to

ϕ

⁢

(

m

)

(here

ϕ

is the

Euler phi function

).

3.

The order of

a

modulo

m

is precisely

ϕ

⁢

(

m

)

if and only if

a

is a

primitive root

for the integer

m

.

Title

multiplicative order of an integer modulo m

Canonical name

MultiplicativeOrderOfAnIntegerModuloM

Date of creation

2013-03-22 16:20:38

Last modified on

2013-03-22 16:20:38

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Definition

Classification

msc 13-00

Classification

msc 13M05

Classification

msc 11-00

Synonym

multiplicative order

Related topic

PrimitiveRoot

Related topic

PrimeResidueClass

Generated on Fri Feb 9 17:19:10 2018 by

LaTeXML
tensor product

tensor product

Summary.

The

tensor product

is a formal

bilinear

multiplication

of two modules or

vector spaces

. In essence, it permits us to replace
bilinear maps from two such objects by an

equivalent

linear map

from
the tensor product of the two objects. The origin of this

operation

lies in classic

differential geometry

and physics, which had need of
multiply indexed geometric objects such as the first and

second
fundamental forms

, and the stress tensor — see Tensor Product (Classical) (

http://planetmath.org/TensorProductClassical

).

Definition (Standard).

Let

R

be a

commutative ring

, and let

A

,

B

be

R

-modules. There exists an

R

-module

A

⊗

B

, called the

tensor product

of

A

and

B

over

R

, together with a canonical
bilinear

homomorphism

⊗

:

A

×

B

→

A

⊗

B

,

distinguished, up to

isomorphism

, by the following

universal
property

.
Every bilinear

R

-

module homomorphism

ϕ

:

A

×

B

→

C

,

lifts to a unique

R

-module homomorphism

ϕ

~

:

A

⊗

B

→

C

,

such that

ϕ

⁢

(

a

,

b

)

=

ϕ

~

⁢

(

a

⊗

b

)

for all

a

∈

A

,

b

∈

B

.

Diagramatically:

\xymatrix

\ar

[

d

r

]

(

.55

)

ϕ

\ar

[

r

]

⊗

A

×

B

&

A

⊗

B

\ar

@

-

-

>

[

d

]

(

.4

)

∃

!

ϕ

~

&

C

The tensor product

A

⊗

B

can be constructed by taking the free

R

-module generated by all formal symbols

a

⊗

b

,

a

∈

A

,

b

∈

B

,

and quotienting by the obvious bilinear

relations

:

(

a

1

+

a

2

)

⊗

b

=

a

1

⊗

b

+

a

2

⊗

b

,

a

1

,

a

2

∈

A

,

b

∈

B

a

⊗

(

b

1

+

b

2

)

=

a

⊗

b

1

+

a

⊗

b

2

,

a

∈

A

,

b

1

,

b

2

∈

B

r

⁢

(

a

⊗

b

)

=

(

r

⁢

a

)

⊗

b

=

a

⊗

(

r

⁢

b

)

a

∈

A

,

b

∈

B

,

r

∈

R

Note.

Basic .

Let

R

be a commutative ring and

L

,

M

,

N

be

R

-modules, then, as modules, we have the following isomorphisms:

1.

R

⊗

M

≅

M

,

2.

M

⊗

N

≅

N

⊗

M

,

3.

(

L

⊗

M

)

⊗

N

≅

L

⊗

(

M

⊗

N

)

4.

(

L

⊕

M

)

⊗

N

≅

(

L

⊗

N

)

⊕

(

M

⊗

N

)

Definition (

Categorical

).

Using the

language

of

categories

, all
of the above can be expressed quite simply by stating that for all

R

-modules

M

, the

functor

(

-

)

⊗

M

is left-adjoint to the
functor

Hom

⁢

(

M

,

-

)

.

Title

tensor product

Canonical name

TensorProduct

Date of creation

2013-03-22 12:21:26

Last modified on

2013-03-22 12:21:26

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

12

Author

rmilson (146)

Entry type

Definition

Classification

msc 13-00

Classification

msc 18-00

Related topic

Module

Related topic

OuterMultiplication

Generated on Fri Feb 9 17:19:16 2018 by

LaTeXML
uniqueness of additive identity in a ring

uniqueness of additive identity in a ring

Lemma 1

.

Let

R

be a ring. There exists a unique element

0

in

R

such that for all

a

in

R

:

0

+

a

=

a

+

0

=

a

.

Proof.

By the

definition

of ring, there exists at least one

identity

in

R

, call it

0

1

. Suppose

0

2

∈

R

is an element which also the of

additive identity

. Thus,

0

2

+

0

1

=

0

2

On the other hand,

0

1

is an additive identity, therefore:

0

2

+

0

1

=

0

1

+

0

2

=

0

1

Hence

0

2

=

0

1

, i.e. there is a unique additive identity.
∎

Title

uniqueness of additive identity in a ring

Canonical

name

UniquenessOfAdditiveIdentityInARing

Date of creation

2013-03-22 14:14:06

Last modified on

2013-03-22 14:14:06

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

6

Author

alozano (2414)

Entry type

Theorem

Classification

msc 13-00

Classification

msc 16-00

Classification

msc 20-00

Generated on Fri Feb 9 17:19:19 2018 by

LaTeXML
unities of ring and subring

unities of ring and subring

Let

R

be a ring and

S

a proper

subring

of it.  Then there exists five cases in all concerning the possible unities of

R

and

S

.

1.

R

and

S

have a common unity.

2.

R

has a unity but

S

does not.

3.

R

and

S

both have their own

non-zero unities

but these are distinct.

4.

R

has no unity but

S

has a non-zero unity.

5.

Neither

R

nor

S

have unity.

Note:

In the cases 3 and 4, the unity of the subring

S

must be a

zero divisor

of

R

.

Examples

1.

The ring

ℚ

and its subring

ℤ

have the common
unity 1.

2.

The subring

S

of

even integers

of the ring

ℤ

has no unity.

3.

Let

S

be the subring of all pairs

(

a

,

0

)

of the ring

R

=

ℤ

×

ℤ

for which the

operations

“

+

” and “

⋅

” are defined componentwise (i.e.

(

a

,

b

)

+

(

c

,

d

)

=

(

a

+

c

,

b

+

d

)

etc.).  Then

S

and

R

have the unities

(

1

,

0

)

and

(

1

,

1

)

, respectively.

4.

Let

S

be the subring of all pairs

(

a

,

0

)

of the ring

R

=

{

(

a

,

2

⁢

b

)

|

a

∈

ℤ

∧

b

∈

ℤ

}

(operations componentwise).  Now

S

has the unity

(

1

,

0

)

but

R

has no unity.

5.

Neither the ring

{

(

2

⁢

a

,

2

⁢

b

)

|

a

,

b

∈

ℤ

}

(operations componentwise) nor its subring consisting of the pairs

(

2

⁢

a

,

0

)

have unity.

Title

unities of ring and subring

Canonical name

UnitiesOfRingAndSubring

Date of creation

2013-03-22 14:49:37

Last modified on

2013-03-22 14:49:37

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

5

Author

pahio (2872)

Entry type

Result

Classification

msc 13-00

Classification

msc 16-00

Classification

msc 20-00

Related topic

UnityOfSubring

Generated on Fri Feb 9 17:19:23 2018 by

LaTeXML
graded ring

graded ring

Let

S

be a groupoid (semigroup,group) and let

R

be a ring (not necessarily with unity) which can be expressed as a

R

=

⊕

s

∈

S

R

s

of

additive

subgroups

R

s

of

R

with

s

∈

S

. If

R

s

⁢

R

t

⊆

R

s

⁢

t

for all

s

,

t

∈

S

then we say that

R

is

groupoid graded

(semigroup-graded, group-graded) ring.

We refer to

R

=

⊕

s

∈

S

R

s

as an

S

-

grading

of

R

and the subgroups

R

s

as the

s

-

components

of

R

. If we have the stronger
condition that

R

s

⁢

R

t

=

R

s

⁢

t

for all

s

,

t

∈

S

, then we say that the ring

R

is

strongly

graded by

S

.

Any element

r

s

in

R

s

(where

s

∈

S

) is said to be

homogeneous of degree

s

. Each element

r

∈

R

can be expressed as a unique and finite sum

r

=

∑

s

∈

S

r

s

of

homogeneous elements

r

s

∈

R

s

.

For any subset

G

⊆

S

we have

R

G

=

∑

g

∈

G

R

g

.
Similarly

r

G

=

∑

g

∈

G

r

g

. If

G

is a subsemigroup of

S

then

R

G

is a

subring

of

R

. If

G

is a left (right, two-sided) ideal of

S

then

R

G

is a left (right, two-sided) ideal of

R

.

Some examples of

graded rings

include:

Polynomial rings

Ring of symmetric functions

Generalised

matrix rings

Morita contexts

Ring of Hirota derivatives

group rings

filtered algebras

Title

graded ring

Canonical name

GradedRing

Date of creation

2013-03-22 11:45:03

Last modified on

2013-03-22 11:45:03

Owner

aplant (12431)

Last modified by

aplant (12431)

Numerical id

19

Author

aplant (12431)

Entry type

Definition

Classification

msc 13A02

Classification

msc 16W30

Classification

msc 14L15

Classification

msc 14L05

Classification

msc 12F10

Classification

msc 11S31

Classification

msc 11S15

Classification

msc 11R33

Synonym

S-graded ring

Synonym

G-graded ring

Related topic

HomogeneousIdeal

Related topic

SupportGradedRing

Defines

groupoid graded ring

Defines

semigroup graded ring

Defines

group graded ring

Defines

homogeneous element

Defines

strongly graded

Generated on Fri Feb 9 17:19:31 2018 by

LaTeXML
homogeneous elements of a graded ring

homogeneous elements of a graded ring

Let

k

be a field, and let

R

be a connected

commutative

k

-

algebra

graded (

http://planetmath.org/GradedAlgebra

) by

ℕ

m

. Then via the

grading

, we can decompose

R

into a

direct sum

of

vector spaces

:

R

=

∐

ω

∈

ℕ

m

R

ω

, where

R

0

=

k

.

For an arbitrary ring element

x

∈

R

, we define the

homogeneous degree

of

x

to be the value

ω

such that

x

∈

R

ω

, and we denote this by

deg

⁡

(

x

)

=

ω

. (See also

homogeneous ideal

)

A set of some importance (ironically), is the

irrelevant ideal

of

R

, denoted by

R

+

, and given by

R

+

=

∐

ω

≠

0

R

ω

.

Finally, we often need to consider the elements of such a ring

R

without using the grading, and we do this by looking at the

homogeneous union

of

R

:

ℋ

⁢

(

R

)

=

⋃

ω

R

ω

.

In particular, in defining a

homogeneous system of parameters

, we are looking at elements of

ℋ

⁢

(

R

+

)

.

References

1

Richard P. Stanley,

Combinatorics and

Commutative Algebra

, Second edition, Birkhauser Press. Boston, MA. 1986.

Title

homogeneous elements

of a

graded ring

Canonical

name

HomogeneousElementsOfAGradedRing

Date of creation

2013-03-22 14:14:52

Last modified on

2013-03-22 14:14:52

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Definition

Classification

msc 13A02

Related topic

HomogeneousIdeal

Defines

homogeneous element

Defines

homogeneous degree

Defines

irrelevant ideal

Defines

homogeneous union

Generated on Fri Feb 9 17:19:34 2018 by

LaTeXML
homogeneous system of parameters

homogeneous system of parameters

Let

k

be a field, let

R

be an

ℕ

m

-graded (

http://planetmath.org/GradedAlgebra

)

k

-

algebra

, and let

M

be a

ℤ

m

-graded

R

-module.

Let

ℋ

⁢

(

R

+

)

be the

homogeneous union

of the irrelevant ideal of

R

.

A

partial homogeneous system of parameters

for

M

is a

finite sequence

of

elements

θ

1

,

θ

2

,

…

,

θ

r

∈

ℋ

⁢

(

R

+

)

such that

dim

⁡

(

M

/

(

∑

i

=

1

r

θ

i

⁢

M

)

)

=

dim

⁡

(

M

)

-

r

,

where

dim

gives the

Krull dimension

.

A ()

homogeneous system of parameters

is a partial homogeneous system of parameters such that

r

=

dim

⁡

(

M

)

.

A sequence

θ

1

,

…

,

θ

r

∈

ℋ

⁢

(

R

+

)

is a

M

-sequence

if for all

i

with

0

≤

i

<

r

, we have that

θ

i

+

1

is not a zero-divisor in

M

/

(

∑

j

=

1

i

θ

i

⁢

M

)

.

Finally, view

M

as being

ℤ

-graded by using any

specialization

of the above

ℤ

m

-

grading

. Then we define the

depth

of

M

to be the length of the longest

homogeneous

M

-sequence.

References

1

Richard P. Stanley,

Combinatorics and

Commutative Algebra

, Second edition, Birkhauser Press. Boston, MA. 1986.

Title

homogeneous system of parameters

Canonical name

HomogeneousSystemOfParameters

Date of creation

2013-03-22 14:14:55

Last modified on

2013-03-22 14:14:55

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Definition

Classification

msc 13A02

Defines

partial homogeneous system of parameters

Defines

complete homogeneous system of parameters

Defines

homogeneous

M

-sequence

Defines

depth

Defines

depth of a module

Generated on Fri Feb 9 17:19:38 2018 by

LaTeXML