aarjaneiro/planetmath huggingface dataset:
example of Klein 4-ring

example of Klein 4-ring

The Klein

4

-ring

K

can be represented by as a

left ideal

of

2

×

2

-matrices
over the field with two elements

ℤ

2

. Doing so helps to explain
some of the unnatural properties of this nonunital ring and is an example of how
many nonunital rings can often be understood as very natural subobjects of

unital
rings

.

K

=

{

[

x

0

y

0

]

:

x

,

y

∈

ℤ

2

}

(1)

To match the product with with the table, use

a

:=

[

1

0

0

0

]

,

b

:=

[

0

0

1

0

]

,

c

:=

[

1

0

1

0

]

.

Here the properties of the abstract multiplication table of

K

(given
here (

http://planetmath.org/Klein4Ring

)) can be
seen as rather natural properties of unital rings. That is, the elements

a

and

c

are

idempotents

in the ring

M

2

⁢

(

ℤ

2

)

and so they
behave similar to

identities

, and

b

is

nilpotent

so that its annihilating
property is expected as well.

The second

noncommutative

nonunital ring of order 4 is the transpose of these matrices,
that is, a

right ideal

of

M

2

⁢

(

ℤ

2

)

.

Viewed in this way we recognize the Klein

4

-ring as part of an infinite family
of similar nonunital rings of left/right ideals of a unital ring. Some authors
prefer to treat such objects only as ideals and not as rings so that the properties
are always given the background of a more familiar structure.

Title

example of

Klein 4-ring

Canonical name

ExampleOfKlein4ring

Date of creation

2013-03-22 17:41:59

Last modified on

2013-03-22 17:41:59

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

9

Author

Algeboy (12884)

Entry type

Example

Classification

msc 16B99

Classification

msc 20-00

Related topic

NonCommutativeRingsOfOrderFour

Generated on Fri Feb 9 18:35:43 2018 by

LaTeXML
isomorphism swapping zero and unity

isomorphism swapping zero and unity

Let

(

R

,

+

,

⋅

)

be a

ring with unity

1.  Define two new

binary operations

of

R

as follows:

a

⊕

b

=

:

a

+

b

-

1

,

a

⊙

b

=

:

a

+

b

-

a

⋅

b

(1)

Then we see that

a

⊕

1

=

a

=

1

⊕

a

,

a

⊙

0

=

a

=

0

⊙

a

.

(2)

But moreover, the

algebraic system

(

R

,

⊕

,

⊙

)

is a unitary ring, too, and

isomorphic

with the original ring.

In fact, we may define the

bijective

mapping

f

:

x

↦

1

-

x

(3)

from

R

to

R

and verify that it is homomorphic:

f

⁢

(

a

)

⊕

f

⁢

(

b

)

=

(

1

-

a

)

⊕

(

1

-

b

)

=

(

1

-

a

)

+

(

1

-

b

)

-

1

=

1

-

a

-

b

=

f

⁢

(

a

+

b

)

,

f

⁢

(

a

)

⊙

f

⁢

(

b

)

=

(

1

-

a

)

⊙

(

1

-

b

)

=

(

1

-

a

)

+

(

1

-

b

)

-

(

1

-

a

)

⋅

(

1

-

b

)

=

1

-

a

⋅

b

=

f

⁢

(

a

⋅

b

)

Thus

(

R

,

⊕

,

⊙

)

as a

homomorphic image

(

http://planetmath.org/HomomorphicImageOfGroup

) of the ring

(

R

,

+

,

⋅

)

is a ring, it’s a question of two isomorphic rings.

Title

isomorphism swapping zero and unity

Canonical name

IsomorphismSwappingZeroAndUnity

Date of creation

2013-03-22 19:17:16

Last modified on

2013-03-22 19:17:16

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Example

Classification

msc 16B99

Classification

msc 20A05

Classification

msc 16S50

Related topic

RingHomomorphism

Related topic

EpimorphismBetweenUnitaryRings

Related topic

Null

Related topic

TranslationAutomorphismOfAPolynomialRing

Generated on Fri Feb 9 18:35:48 2018 by

LaTeXML
opposite ring

opposite ring

If

R

is a ring, then we may construct the

opposite ring

R

o

⁢

p

which has the same underlying

abelian group

structure

, but with

multiplication

in the opposite order: the

product

of

r

1

and

r

2

in

R

o

⁢

p

is

r

2

⁢

r

1

.

If

M

is a left

R

-module, then it can be made into a right

R

o

⁢

p

-module, where a module

element

m

, when multiplied on the right by an element

r

of

R

o

⁢

p

, yields the

r

⁢

m

that we have with our left

R

-

module action

on

M

. Similarly, right

R

-modules can be made into left

R

o

⁢

p

-modules.

If

R

is a

commutative ring

, then it is equal to its own opposite ring.

Similar

constructions

occur in

the opposite group and opposite category.

Title

opposite ring

Canonical name

OppositeRing

Date of creation

2013-03-22 11:51:14

Last modified on

2013-03-22 11:51:14

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

7

Author

antizeus (11)

Entry type

Definition

Classification

msc 16B99

Classification

msc 17A01

Related topic

DualCategory

Related topic

NonCommutativeRingsOfOrderFour

Generated on Fri Feb 9 18:35:49 2018 by

LaTeXML
surjective homomorphism between unitary rings

surjective homomorphism between unitary rings

Theorem

.

Let

f

be a

surjective

homomorphism

from a

unitary ring

R

to another unitary ring

R

′

.  Then

•

f

⁢

(

1

)

=

1

′

,

•

f

⁢

(

a

-

1

)

=

(

f

⁢

(

a

)

)

-

1

for all

elements

a

belonging to the

group of units

of

R

.

Proof.

1

∘

.  In a ring, the

identity element is unique

, whence it suffices to show that

f

⁢

(

1

)

has the properties required for the unity of the ring

R

′

.  When

a

′

is an arbitrary element of this ring, there is by the surjectivity an element

a

of

R

such that

f

⁢

(

a

)

=

a

′

.  Thus we have

f

⁢

(

1

)

⁢

a

′

=

f

⁢

(

1

)

⁢

f

⁢

(

a

)

=

f

⁢

(

1

⁢

a

)

=

f

⁢

(

a

)

=

a

′

,

a

′

⁢

f

⁢

(

1

)

=

f

⁢

(

a

)

⁢

f

⁢

(

1

)

=

f

⁢

(

a

⁢

1

)

=

f

⁢

(

a

)

=

a

′

.

2

∘

.  Let

a

be a unit of

R

.  Then

f

⁢

(

a

)

⁢

f

⁢

(

a

-

1

)

=

f

⁢

(

a

⁢

a

-

1

)

=

f

⁢

(

1

)

=

1

′

,

f

⁢

(

a

-

1

)

⁢

f

⁢

(

a

)

=

f

⁢

(

a

-

1

⁢

a

)

=

f

⁢

(

1

)

=

1

′

,

whence

f

⁢

(

a

-

1

)

is a

multiplicative inverse

of

f

⁢

(

a

)

.

Title

surjective homomorphism between unitary rings

Canonical name

SurjectiveHomomorphismBetweenUnitaryRings

Date of creation

2013-03-22 19:10:22

Last modified on

2013-03-22 19:10:22

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Theorem

Classification

msc 16B99

Classification

msc 13B10

Related topic

IsomorphismSwappingZeroAndUnity

Generated on Fri Feb 9 18:35:51 2018 by

LaTeXML
unit

unit

Let

R

be a ring with

multiplicative identity

1

. We say that

u

∈

R

is an unit (or unital) if

u

divides

1

(denoted

u

∣

1

). That is, there exists an

r

∈

R

such that

1

=

u

⁢

r

=

r

⁢

u

.

Notice that

r

will be the

multiplicative inverse

(in the ring) of

u

, so we can characterize the units as those elements of the ring having multiplicative inverses.

In the special case that

R

is the

ring of integers

of an

algebraic number field

K

, the units of

R

are sometimes called the

algebraic units

of

K

(and also the units of

K

).  They are determined by Dirichlet’s unit

theorem

.

Title

unit

Canonical

name

Unit

Date of creation

2013-03-22 11:56:28

Last modified on

2013-03-22 11:56:28

Owner

drini (3)

Last modified by

drini (3)

Numerical id

15

Author

drini (3)

Entry type

Definition

Classification

msc 16B99

Synonym

unital

Related topic

Associates

Related topic

Prime

Related topic

Ring

Related topic

UnitsOfQuadraticFields

Defines

algebraic unit

Generated on Fri Feb 9 18:35:53 2018 by

LaTeXML
annihilator

annihilator

Let

R

be a ring, and suppose that

M

is a left

R

-module and

N

a right

R

-module.

Annihilator of a Subset of a Module

1.

If

X

is a subset of

M

,
then we define the

left annihilator

of

X

in

R

:

l

.

ann

⁢

(

X

)

=

{

r

∈

R

∣

r

⁢

x

=

0

⁢

for all

⁢

x

∈

X

}

.

If

a

,

b

∈

l

.

ann

⁢

(

X

)

, then so are

a

-

b

and

r

⁢

a

for all

r

∈

R

. Therefore,

l

.

ann

⁢

(

X

)

is a

left ideal

of

R

.

2.

If

Y

is a subset of

N

,
then we define the

right annihilator

of

Y

in

R

:

r

.

ann

⁢

(

Y

)

=

{

r

∈

R

∣

y

⁢

r

=

0

⁢

for all

⁢

y

∈

Y

}

.

Like above, it is easy to see that

r

.

ann

⁢

(

Y

)

is a

right ideal

of

R

.

Remark

.

l

.

ann

⁢

(

X

)

and

r

.

ann

⁢

(

Y

)

may also be written as

l

.

ann

R

⁢

(

X

)

and

r

.

ann

R

⁢

(

Y

)

respectively, if we want to emphasize

R

.

Annihilator of a Subset of a Ring

1.

If

Z

is a subset of

R

,
then we define the

right annihilator

of

Z

in

M

:

r

.

ann

M

⁢

(

Z

)

=

{

m

∈

M

∣

z

⁢

m

=

0

⁢

for all

⁢

z

∈

Z

}

.

If

m

,

n

∈

r

.

ann

M

⁢

(

Z

)

, then so are

m

-

n

and

r

⁢

m

for all

r

∈

R

. Therefore,

r

.

ann

M

⁢

(

Z

)

is a left

R

-

submodule

of

M

.

2.

If

Z

is a subset of

R

,
then we define the

left annihilator

of

Z

in

N

:

l

.

ann

N

⁢

(

Z

)

=

{

n

∈

N

∣

n

⁢

z

=

0

⁢

for all

⁢

z

∈

Z

}

.

Similarly, it can be easily seen that

l

.

ann

N

⁢

(

Z

)

is a right

R

-submodule of

N

.

Title

annihilator

Canonical name

Annihilator

Date of creation

2013-03-22 12:01:32

Last modified on

2013-03-22 12:01:32

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

8

Author

antizeus (11)

Entry type

Definition

Classification

msc 16D10

Synonym

left annihilator

Synonym

right annihilator

Related topic

JacobsonRadical

Generated on Fri Feb 9 18:35:55 2018 by

LaTeXML
annihilator is an ideal

annihilator is an ideal

The

right annihilator

of a right

R

-module

M

R

in

R

is an ideal.

Proof:

By the

distributive law

for modules, it is easy to see that

r

.

ann

⁡

(

M

R

)

is

closed under

addition

and right multiplication.
Now take

x

∈

r

.

ann

⁡

(

M

R

)

and

r

∈

R

.

Take any

m

∈

M

R

. Then

m

⁢

r

∈

M

R

, but then

(

m

⁢

r

)

⁢

x

=

0

since

x

∈

r

.

ann

⁡

(

M

R

)

. So

m

⁢

(

r

⁢

x

)

=

0

and

r

⁢

x

∈

r

.

ann

⁡

(

M

R

)

.

An

equivalent

result holds for left annihilators.

Title

annihilator

is an ideal

Canonical name

AnnihilatorIsAnIdeal

Date of creation

2013-03-22 12:50:27

Last modified on

2013-03-22 12:50:27

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

10

Author

yark (2760)

Entry type

Theorem

Classification

msc 16D10

Classification

msc 16D25

Generated on Fri Feb 9 18:35:57 2018 by

LaTeXML
artinian

artinian

A module

M

is

artinian

if it

satisfies

the following

equivalent

conditions:

•

the

descending chain condition

holds for

submodules

of

M

;

•

every nonempty family of submodules of

M

has a

minimal element

.

A ring

R

is

left artinian

if it is artinian as a

left module

over itself
(i.e. if

R

R

is an

artinian module

),
and

right artinian

if it is artinian as a right module over itself
(i.e. if

R

R

is an artinian module),
and simply

artinian

if both conditions hold.

Title

artinian

Canonical name

Artinian

Date of creation

2013-03-22 12:26:46

Last modified on

2013-03-22 12:26:46

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

6

Author

antizeus (11)

Entry type

Definition

Classification

msc 16D10

Synonym

left artinian

Synonym

right artinian

Related topic

Noetherian

Related topic

Noetherian2

Related topic

HollowMatrixRings

Generated on Fri Feb 9 18:35:58 2018 by

LaTeXML
chain conditions in vector spaces

chain conditions in vector spaces

From the

theorem

in the parent article - that an

A

-module

M

has a

composition series

if and only if it

satisfies

both

chain conditions

- it is easy to see that

Theorem 1

.

Let

k

be a field,

V

a

k

-

vector space

. Then the following are

equivalent

:

1.

V

is

finite-dimensional

;

2.

V

has a composition series;

3.

V

satisfies the

ascending chain condition

(acc);

4.

V

satisfies the

descending chain condition

(dcc).

Proof.

Clearly (1)

⇒

(2), since

submodules

are just

subspaces

. (2)

⇒

(3) and (2)

⇒

(4) from the parent article. So it remains to see that (3)

⇒

(1) and (4)

⇒

(1). But if

V

is infinite-dimensional, we can choose a

sequence

{

x

i

}

i

≥

1

of

linearly independent

elements

. Let

U

n

be the subspace spanned by

x

1

,

…

,

x

n

and

V

n

the subspace spanned by

x

n

+

1

,

x

n

+

2

,

…

. Then the

U

i

form a strictly ascending

infinite

family of subspaces, so

V

does not satisfy the ascending chain condition; the

V

i

form a strictly descending infinite family of subspaces, so

V

does not satisfy the descending chain condition.
∎

This easily implies the following:

Corollary 1

.

Let

A

be a ring in which

(

0

)

=

m

1

⁢

…

⁢

m

n

where the

m

i

are (not necessarily distinct)

maximal ideals

. Then

A

is

Noetherian

if and only if

A

is

Artinian

.

Proof.

We have the sequence of ideals

A

⊃

𝔪

1

⊃

𝔪

1

⁢

𝔪

2

⊃

…

⊃

𝔪

1

⁢

…

⁢

𝔪

n

=

0

Each factor

𝔪

1

⁢

…

⁢

𝔪

i

-

1

/

𝔪

1

⁢

…

⁢

𝔪

i

is a vector space over the field

A

/

𝔪

i

. By the above theorem, each

quotient

satisfies the acc if and only if it satisfies the dcc. But by repeatedly applying the fact that in a

short exact sequence

, the middle term satisfies the acc (dcc) if and only if both ends do, we see that

A

satisfies the acc if and only if it satisfies the dcc.
∎

References

1

M.F. Atiyah, I.G. MacDonald,

Introduction to Commutative Algebra

, Addison-Wesley 1969.

Title

chain conditions in vector spaces

Canonical name

ChainConditionsInVectorSpaces

Date of creation

2013-03-22 19:11:55

Last modified on

2013-03-22 19:11:55

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

4

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 16D10

Generated on Fri Feb 9 18:36:00 2018 by

LaTeXML
composition series

composition series

Let

R

be a ring and let

M

be a (right or left)

R

-module.
A series of

submodules

M

=

M

0

⊃

M

1

⊃

M

2

⊃

…

⊃

M

n

=

0

in which each

quotient

M

i

/

M

i

+

1

is simple is called a

composition series

for

M

.

A module need not have a composition series. For example, the

ring of integers

,

ℤ

, considered as a module over itself, does not have a composition series.

A necessary and

sufficient

condition for a module to have a composition series is that it is both

Noetherian

and

Artinian

.

If a module does have a composition series, then all composition series are the same length.
This length (the number

n

above) is called the

composition

length

of the module.

If

R

is a

semisimple

Artinian ring, then

R

R

and

R

R

always have composition series.

Title

composition series

Canonical name

CompositionSeries

Date of creation

2013-03-22 14:04:13

Last modified on

2013-03-22 14:04:13

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

6

Author

mclase (549)

Entry type

Definition

Classification

msc 16D10

Generated on Fri Feb 9 18:36:02 2018 by

LaTeXML
conjugate module

conjugate module

If

M

is a

right module

over a ring

R

,
and

α

is an

endomorphism

of

R

,
we define the

conjugate module

M

α

to be the right

R

-module
whose underlying set is

{

m

α

∣

m

∈

M

}

,
with

abelian group

structure

identical to that of

M

(i.e.

(

m

-

n

)

α

=

m

α

-

n

α

),
and scalar multiplication given by

m

α

⋅

r

=

(

m

⋅

α

⁢

(

r

)

)

α

for all

m

in

M

and

r

in

R

.

In other words, if

ϕ

:

R

→

End

ℤ

⁢

(

M

)

is the

ring homomorphism

that describes
the

right module action

of

R

upon

M

,
then

ϕ

⁢

α

describes
the right module action of

R

upon

M

α

.

If

N

is a left

R

-module, we define

N

α

similarly,
with

r

⋅

n

α

=

(

α

(

r

)

⋅

n

)

α

.

Title

conjugate module

Canonical name

ConjugateModule

Date of creation

2013-03-22 11:49:47

Last modified on

2013-03-22 11:49:47

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

9

Author

antizeus (11)

Entry type

Definition

Classification

msc 16D10

Classification

msc 41A45

Generated on Fri Feb 9 18:36:03 2018 by

LaTeXML
direct product of modules

direct product of modules

Let

{

X

i

:

i

∈

I

}

be a

collection

of modules
in some

category

of modules.
Then the

direct product

∏

i

∈

I

X

i

of that collection is the module
whose underlying set is the

Cartesian product

of the

X

i

with componentwise

addition

and

scalar multiplication

.
For example, in a category of

left modules

:

(

x

i

)

+

(

y

i

)

=

(

x

i

+

y

i

)

,

r

⁢

(

x

i

)

=

(

r

⁢

x

i

)

.

For each

j

∈

I

we have
a

projection

p

j

:

∏

i

∈

I

X

i

→

X

j

defined by

(

x

i

)

↦

x

j

,
and
an

injection

λ

j

:

X

j

→

∏

i

∈

I

X

i

where an

element

x

j

of

X

j

maps to the

element of

∏

i

∈

I

X

i

whose

j

th term is

x

j

and every other term is zero.

The direct product

∏

i

∈

I

X

i

satisfies

a certain

universal property

.
Namely, if

Y

is a module
and there exist

homomorphisms

f

i

:

X

i

→

Y

for all

i

∈

I

,
then there exists a unique homomorphism

ϕ

:

Y

→

∏

i

∈

I

X

i

satisfying

ϕ

⁢

λ

i

=

f

i

for all

i

∈

I

.

\xymatrix

X

i

\ar

[

d

r

]

λ

i

\ar

[

r

r

]

f

i

&

&

Y

\ar

@

-

-

>

[

d

l

]

ϕ

&

∏

i

∈

I

X

i

The direct product is often referred to
as the

complete direct sum

,
or the

strong direct sum

,
or simply the .

Compare this to the direct sum of modules.

Title

direct product of modules

Canonical name

DirectProductOfModules

Date of creation

2013-03-22 12:09:34

Last modified on

2013-03-22 12:09:34

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

10

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 16D10

Synonym

strong direct sum

Synonym

complete direct sum

Related topic

CategoricalDirectProduct

Defines

direct product

Generated on Fri Feb 9 18:36:05 2018 by

LaTeXML
direct summand

direct summand

Let

R

be a ring and

B

⊆

A

left (right)

R

-modules. Then

B

is called a

direct summand

of

A

if there exists a left (right)

R

-

submodule

C

such that

A

=

B

⊕

C

.

For example, a

projective module

is a direct summand of a

free module

over any ring.

Title

direct summand

Canonical name

DirectSummand

Date of creation

2013-03-22 14:51:42

Last modified on

2013-03-22 14:51:42

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16D10

Related topic

DirectSum

Generated on Fri Feb 9 18:36:07 2018 by

LaTeXML
example of an Artinian module which is not Noetherian

example of an Artinian module which is not Noetherian

It is well known, that left (right)

Artinian ring

is left (right)

Noetherian

(Akizuki-Hopkins-Levitzki theorem). We will show that this no longer holds for modules.

Let

ℤ

be the

ring of integers

and

ℚ

the field of rationals. Let

p

∈

ℤ

be a

prime number

and consider

G

=

{

a

p

n

∈

ℚ

|

a

∈

ℤ

;

n

≥

0

}

.

Of course

G

is a

ℤ

-module via standard

multiplication

and

addition

. For

n

≥

0

consider

G

n

=

{

a

p

n

∈

ℚ

|

a

∈

ℤ

}

.

Of course each

G

n

⊆

G

is a

submodule

and it is easy to see, that

ℤ

=

G

0

⊂

G

1

⊂

G

2

⊂

G

3

⊂

⋯

,

where each

inclusion

is proper. We will show that

G

/

ℤ

is

Artinian

, but it is not Noetherian.

Let

π

:

G

→

G

/

ℤ

be the

canonical projection

. Then

G

n

′

=

π

⁢

(

G

n

)

is a submodule of

G

/

ℤ

and

0

=

G

0

′

⊂

G

1

′

⊂

G

2

′

⊂

G

3

′

⊂

G

4

′

⊂

⋯

.

The inclusions are proper, because for any

n

>

0

we have

G

n

+

1

′

/

G

n

′

≃

(

G

n

+

1

/

ℤ

)

/

(

G

n

/

ℤ

)

≃

G

n

+

1

/

G

n

≠

0

,

due to

Third Isomorphism Theorem

for modules. This shows, that

G

/

ℤ

is not Noetherian.

In order to show that

G

/

ℤ

is Artinian, we will show, that each

proper submodule

of

G

/

ℤ

is of the form

G

n

′

. Let

N

⊆

G

/

ℤ

be a proper submodule. Assume that for some

a

∈

ℤ

and

n

≥

0

we have

a

p

n

+

ℤ

∈

N

.

We may assume that

gcd

⁢

(

a

,

p

n

)

=

1

. Therefore there are

α

,

β

∈

ℤ

such that

1

=

α

⁢

a

+

β

⁢

p

n

.

Now, since

N

is a

ℤ

-module we have

α

⁢

a

p

n

+

ℤ

∈

N

and since

0

+

ℤ

=

β

+

ℤ

=

β

⁢

p

n

p

n

+

ℤ

∈

N

we have that

1

p

n

+

ℤ

=

α

⁢

a

+

β

⁢

p

n

p

n

+

ℤ

∈

N

.

Now, let

m

>

0

be the smallest number, such that

1

p

m

+

ℤ

∉

N

. What we showed is that

N

=

G

m

-

1

′

=

π

⁢

(

G

m

-

1

)

,

because for every

0

≤

n

≤

m

-

1

(and only for such

n

) we have

1

p

n

+

ℤ

∈

N

and thus

N

is a image of a submodule of

G

, which is generated by

1

p

n

and this is precisely

G

m

-

1

. Now let

N

1

⊇

N

2

⊇

N

3

⊇

⋯

be a chain of submodules in

G

/

ℤ

. Then there are

natural numbers

n

1

,

n

2

,

…

such that

N

i

=

G

n

i

′

. Note that

G

k

′

⊇

G

s

′

if and only if

k

≥

s

. In particular we obtain a

sequence

of natural numbers

n

1

≥

n

2

≥

n

3

≥

⋯

This chain has to stabilize, which

completes

the proof.

□

Title

example of an

Artinian module

which is not Noetherian

Canonical name

ExampleOfAnArtinianModuleWhichIsNotNoetherian

Date of creation

2013-03-22 19:04:18

Last modified on

2013-03-22 19:04:18

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

5

Author

joking (16130)

Entry type

Example

Classification

msc 16D10

Generated on Fri Feb 9 18:36:09 2018 by

LaTeXML
finitely generated module

finitely generated module

A module

X

over a ring

R

is said to be

finitely generated

if there is a finite subset

Y

of

X

such that

Y

spans

X

. Let us recall that the span of a (not necessarily finite) set

X

of vectors is the class of all (finite)

linear combinations

of elements of

S

; moreover, let us recall that the span of the

empty set

is defined to be the singleton consisting of only one vector, the

zero vector

0

→

. A module

X

is then called

cyclic

if it can be a singleton.

Examples

. Let

R

be a

commutative ring

with 1 and

x

be an indeterminate.

1.

R

⁢

x

=

{

r

⁢

x

∣

r

∈

R

}

is a cyclic

R

-module generated by

{

x

}

.

2.

R

⊕

R

⁢

x

is a finitely-generated

R

-module generated by

{

1

,

x

}

. Any

element

in

R

⊕

R

⁢

x

can be expressed uniquely as

r

+

s

⁢

x

.

3.

R

⁢

[

x

]

is not finitely generated as an

R

-module. For if there is a

finite set

Y

R

⁢

[

x

]

, taking

d

to be the largest of all degrees of polynomials in

Y

, then

x

d

+

1

would not be in the of

Y

, assumed to be

R

⁢

[

x

]

, which is a

contradiction

. (Note, however, that

R

⁢

[

x

]

is finitely-generated as an

R

-

algebra

.)

Title

finitely generated module

Canonical name

FinitelyGeneratedModule

Date of creation

2013-03-22 14:01:08

Last modified on

2013-03-22 14:01:08

Owner

Thomas Heye (1234)

Last modified by

Thomas Heye (1234)

Numerical id

15

Author

Thomas Heye (1234)

Entry type

Definition

Classification

msc 16D10

Related topic

ModuleFinite

Related topic

Span

Defines

finitely generated

Defines

cyclic module

Generated on Fri Feb 9 18:36:11 2018 by

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