aarjaneiro/planetmath huggingface dataset:
standard identity

standard identity

Let

R

be a

commutative ring

and

X

be a set of non-commuting

variables

over

R

. The

standard identity

of degree

n

in

R

⁢

⟨

X

⟩

, denoted by

[

x

1

,

…

⁢

x

n

]

, is the

polynomial

∑

π

sign

⁡

(

π

)

⁢

x

π

⁢

(

1

)

⁢

⋯

⁢

x

π

⁢

(

n

)

,

where

⁢

π

∈

S

n

.

Remarks:

•

A ring

R

satisfying the standard identity of degree 2 (i.e.,

[

R

,

R

]

=

0

) is

commutative

. In this sense,

algebras

satisfying a standard identity is a

generalization

of the class of commutative algebras.

•

Two immediate

properties

of

[

x

1

,

…

⁢

x

n

]

are that it is

multilinear

over

R

, and it is

alternating

, in the sense that

[

r

1

,

…

⁢

r

n

]

=

0

whenever two of the

r

i

′

⁢

s

are equal. Because of these two properties, one can show that an n-dimensional algebra

R

over a field

k

is a

PI-algebra

, satisfying the standard identity of degree

n

+

1

. As a

corollary

,

𝕄

n

⁢

(

k

)

, the

n

×

n

matrix ring

over a field

k

, is a PI-algebra satisfying the standard identity of degree

n

2

+

1

. In fact, Amitsur and Levitski have shown that

𝕄

n

⁢

(

k

)

actually

satisfies

the standard identity of degree

2

⁢

n

.

References

1

S. A. Amitsur and J. Levitski,

Minimal

identities

for algebras

, Proc. Amer. Math. Soc., 1 (1950) 449-463.

Title

standard identity

Canonical

name

StandardIdentity

Date of creation

2013-03-22 14:21:10

Last modified on

2013-03-22 14:21:10

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16R10

Generated on Fri Feb 9 18:42:27 2018 by

LaTeXML
T-ideal

T-ideal

Let

R

be a

commutative ring

and

R

⁢

⟨

X

⟩

be a

free algebra

over

R

on a set

X

of

non-commuting

variables

. A

two-sided ideal

I

of

R

⁢

⟨

X

⟩

is called a

T

-

ideal

if

ϕ

⁢

(

I

)

⊆

I

for any

R

-

endomorphism

ϕ

of

R

⁢

⟨

X

⟩

.

For example, let

A

be a

R

-

algebra

. Define

𝒯

⁢

(

A

)

to be the set of all

polynomial identities

(

http://planetmath.org/PolynomialIdentityAlgebra

)

f

∈

R

⁢

⟨

X

⟩

for

A

. Then

𝒯

⁢

(

A

)

is a

T

-ideal of

R

⁢

⟨

X

⟩

.

𝒯

⁢

(

A

)

is called the

T

-

ideal of of A

.

Title

T-ideal

Canonical name

Tideal

Date of creation

2013-03-22 14:21:12

Last modified on

2013-03-22 14:21:12

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16R10

Generated on Fri Feb 9 18:42:28 2018 by

LaTeXML
polarization by differential operators

polarization by differential operators

One can construct the polars of a

polynomial

by means of a differential
operator. Suppose we have a

homogeneous polynomial

p

⁢

(

x

1

,

…

,

x

n

)

.
To compute the polars of

p

we act on it with the operator

Δ

=

y

1

⁢

∂

/

∂

⁡

x

1

+

⋯

+

y

n

⁢

∂

/

∂

⁡

x

n

;  the

k

-th polar of

p

equals

Δ

k

⁢

p

⁢

(

x

1

,

…

⁢

x

n

)

.

Title

polarization by differential operators

Canonical name

PolarizationByDifferentialOperators

Date of creation

2013-03-22 17:37:26

Last modified on

2013-03-22 17:37:26

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 16R99

Classification

msc 15A69

Classification

msc 15A63

Classification

msc 17A99

Generated on Fri Feb 9 18:42:33 2018 by

LaTeXML
Ore domain

Ore domain

Let

R

be a domain (

http://planetmath.org/IntegralDomain

).
We say that

R

is a

right Ore domain

if any two nonzero elements of

R

have a nonzero common right

multiple

,
i.e. for every pair of nonzero

x

and

y

,
there exists a pair of

elements

r

and

s

of

R

such that

x

⁢

r

=

y

⁢

s

≠

0

.

This condition turns out to be

equivalent

to the following conditions on

R

when viewed as a right

R

-module:

(a)

R

R

is a

uniform module

.

(b)

R

R

is a

module of finite rank

.

The definition of a

left

Ore domain

is similar.

If

R

is a

commutative

domain (

http://planetmath.org/IntegralDomain

),
then it is a right (and left) Ore domain.

Title

Ore domain

Canonical name

OreDomain

Date of creation

2013-03-22 11:51:17

Last modified on

2013-03-22 11:51:17

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

11

Author

antizeus (11)

Entry type

Definition

Classification

msc 16S10

Generated on Fri Feb 9 18:42:34 2018 by

LaTeXML
Friedrichs’ theorem

Friedrichs’ theorem

Fix a

commutative

unital ring

K

of

characteristic

0. Let

X

be a finite
set and

K

⁢

⟨

X

⟩

the

free associative algebra

on

X

. Then define
the map

δ

:

K

⁢

⟨

X

⟩

→

K

⁢

⟨

X

⟩

⊗

K

⁢

⟨

X

⟩

by

x

↦

x

⊗

1

+

1

⊗

x

.

Theorem 1

(Friedrichs).

[

1

, Thm V.9]

An

element

a

∈

K

⁢

⟨

X

⟩

is a

Lie element

if and only if

a

⁢

δ

=

a

⊗

1

+

1

⊗

a

.

The term Lie element applies only when an element is taken from the

universal
enveloping algebra

of a

Lie algebra

. Here the Lie algebra in question is
the

free Lie algebra

on

X

,

F

⁢

L

⁢

⟨

X

⟩

whose universal enveloping
algebra is

K

⁢

⟨

X

⟩

by a theorem of Witt.

This characterization of Lie elements is a primary means in modern proofs
of the Baker-Campbell-Hausdorff formula.

References

1

Nathan Jacobson

Lie Algebras

, Interscience Publishers, New York, 1962.

Title

Friedrichs’ theorem

Canonical name

FriedrichsTheorem

Date of creation

2013-03-22 16:51:16

Last modified on

2013-03-22 16:51:16

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

6

Author

Algeboy (12884)

Entry type

Theorem

Classification

msc 16S30

Classification

msc 17B35

Generated on Fri Feb 9 18:42:39 2018 by

LaTeXML
orthogonal idempotents of the group ring

orthogonal idempotents of the group ring

Let

G

be a finite abelian group, let

L

be any field containing the

|

G

|

-th

roots of unity

, and let

G

^

denote the character group of

G

with values in

L

. For any

character

χ

∈

G

^

, we define

ε

χ

, the corresponding

orthogonal idempotent

of the

group ring

L

⁢

[

G

]

, by

ε

χ

=

1

|

G

|

⁢

∑

g

∈

G

χ

⁢

(

g

)

⁢

g

-

1

.

The following equalities hold:

•

ε

χ

2

=

ε

χ

for all

χ

•

ε

χ

⁢

ε

ψ

=

0

for any

χ

≠

ψ

•

∑

χ

∈

G

^

ε

χ

=

1

•

ε

χ

⋅

g

=

χ

⁢

(

g

)

⁢

ε

χ

These orthogonal idempotents are used to decompose modules over

L

⁢

[

G

]

: If

M

is such a module, then

M

=

⊕

χ

(

ε

χ

⁢

M

)

.

Title

orthogonal idempotents of the group ring

Canonical name

OrthogonalIdempotentsOfTheGroupRing

Date of creation

2013-03-22 14:12:42

Last modified on

2013-03-22 14:12:42

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Definition

Classification

msc 16S34

Generated on Fri Feb 9 18:42:41 2018 by

LaTeXML
support

support

Let

R

⁢

[

G

]

be the

group ring

of a group

G

over a ring

R

.

Let

x

=

∑

g

x

g

⁢

g

be an element of

R

⁢

[

G

]

. The

support

of

x

, often written

supp

⁡

(

x

)

, is the set of elements of

G

which occur with non-zero coefficient in the expansion of

x

.

Thus:

supp

⁡

(

x

)

=

{

g

∈

G

∣

x

g

≠

0

}

.

Title

support

Canonical name

Support12

Date of creation

2013-03-22 14:02:56

Last modified on

2013-03-22 14:02:56

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

5

Author

mclase (549)

Entry type

Definition

Classification

msc 16S34

Generated on Fri Feb 9 18:42:43 2018 by

LaTeXML
q skew derivation

q skew derivation

Let

(

σ

,

δ

)

be a

skew derivation

on a ring

R

.
Let

q

be a

central

(

σ

,

δ

)

-constant (

http://planetmath.org/SigmaDeltaConstant

).
Suppose further that

δ

⁢

σ

=

q

⋅

σ

⁢

δ

.
Then we say that

(

σ

,

δ

)

is a

q

-.

Title

q skew derivation

Canonical name

QSkewDerivation

Date of creation

2013-03-22 11:49:41

Last modified on

2013-03-22 11:49:41

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

10

Author

antizeus (11)

Entry type

Definition

Classification

msc 16S36

Related topic

QSkewPolynomialRing

Generated on Fri Feb 9 18:42:44 2018 by

LaTeXML
q skew polynomial ring

q

skew polynomial ring

If

(

σ

,

δ

)

is a

q

-

skew derivation

(

http://planetmath.org/QSkewDerivation

) on

R

,
then we say that the

skew polynomial ring

R

⁢

[

θ

;

σ

,

δ

]

is a

q

-skew polynomial ring

.

Title

q

skew polynomial ring

Canonical

name

QSkewPolynomialRing

Date of creation

2013-03-22 11:49:44

Last modified on

2013-03-22 11:49:44

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

9

Author

antizeus (11)

Entry type

Definition

Classification

msc 16S36

Classification

msc 65Q05

Related topic

QSkewDerivation

Generated on Fri Feb 9 18:42:46 2018 by

LaTeXML
sigma, delta constant

sigma, delta constant

If

(

σ

,

δ

)

is a

skew derivation

on a ring

R

,
then a

(

σ

,

δ

)

-constant

is an element

q

of

R

such that

σ

⁢

(

q

)

=

q

and

δ

⁢

(

q

)

=

0

.

Note: If

q

is a

(

σ

,

δ

)

-constant,
then it follows that

σ

⁢

(

q

⋅

x

)

=

q

⋅

σ

⁢

(

x

)

and

δ

⁢

(

q

⋅

x

)

=

q

⋅

δ

⁢

(

x

)

for all

x

in

R

.

Title

sigma, delta constant

Canonical name

SigmaDeltaConstant

Date of creation

2013-03-22 11:49:38

Last modified on

2013-03-22 11:49:38

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

8

Author

antizeus (11)

Entry type

Definition

Classification

msc 16S36

Generated on Fri Feb 9 18:42:48 2018 by

LaTeXML
sigma derivation

sigma derivation

If

σ

is a ring

endomorphism

on a ring

R

,
then a

(left)

σ

-

derivation

is an

additive

map

δ

on

R

such that

δ

⁢

(

x

⋅

y

)

=

σ

⁢

(

x

)

⋅

δ

⁢

(

y

)

+

δ

⁢

(

x

)

⋅

y

for all

x

,

y

in

R

.

Title

sigma derivation

Canonical name

SigmaDerivation

Date of creation

2013-03-22 11:42:50

Last modified on

2013-03-22 11:42:50

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

22

Author

antizeus (11)

Entry type

Definition

Classification

msc 16S36

Classification

msc 81S40

Classification

msc 81Txx

Classification

msc 18E05

Classification

msc 55N40

Classification

msc 18-00

Generated on Fri Feb 9 18:42:50 2018 by

LaTeXML
skew derivation

skew derivation

A

(left)

skew derivation

on a ring

R

is a pair

(

σ

,

δ

)

,
where

σ

is a ring

endomorphism

of

R

,
and

δ

is a left

σ

-

derivation

(

http://planetmath.org/SigmaDerivation

) on

R

.

Title

skew derivation

Canonical name

SkewDerivation

Date of creation

2013-03-22 11:49:21

Last modified on

2013-03-22 11:49:21

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

9

Author

antizeus (11)

Entry type

Definition

Classification

msc 16S36

Related topic

SigmaDerivation

Generated on Fri Feb 9 18:42:51 2018 by

LaTeXML
skew polynomial ring

skew polynomial ring

If

(

σ

,

δ

)

is a left

skew derivation

on

R

,
then we can construct the

(left)

skew polynomial ring

R

⁢

[

θ

;

σ

,

δ

]

, which is made up of

polynomials

in an

indeterminate

θ

and left-hand coefficients from

R

,
with

multiplication

satisfying the

relation

θ

⋅

r

=

σ

⁢

(

r

)

⋅

θ

+

δ

⁢

(

r

)

for all

r

in

R

.

Title

skew polynomial ring

Canonical

name

SkewPolynomialRing

Date of creation

2013-03-22 11:49:35

Last modified on

2013-03-22 11:49:35

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

7

Author

antizeus (11)

Entry type

Definition

Classification

msc 16S36

Generated on Fri Feb 9 18:42:53 2018 by

LaTeXML
Weyl algebra

Weyl algebra

Abstract definition

Let

F

be a field and

V

be an

F

-

vector space

with basis

{

P

i

}

i

∈

I

∪

{

Q

i

}

i

∈

I

, where

I

is some non-empty

index set

. Let

T

be the

tensor algebra

of

V

and let

J

be the ideal in

T

generated by the set

{

P

i

⊗

Q

j

-

Q

j

⊗

P

i

-

δ

i

⁢

j

}

i

,

j

∈

I

where

δ

is the

Kronecker delta

symbol. Then the

quotient

T

/

J

is the

|

I

|

-th

Weyl algebra

.

A more concrete definition

If the field

F

has characteristic zero we have the following more
concrete definition. Let

R

:=

F

⁢

[

{

X

i

}

i

∈

I

]

be the

polynomial
ring

over

F

in

indeterminates

X

i

labeled by

I

. For any

i

∈

I

, let

∂

i

denote the partial differential

operator

with
respect to

X

i

. Then the

|

I

|

-th Weyl algebra is the set

W

of all
differential operators of the form

D

=

∑

|

α

|

≤

n

f

α

⁢

∂

α

where the summation

variable

α

is a multi-index with

|

I

|

entries,

n

is the degree of

D

, and

f

α

∈

R

. The

algebra

structure

is defined by the usual operator

multiplication

, where the

coefficients

f

α

∈

R

are identified with the operators of left
multiplication with them for conciseness of notation. Since the

derivative

of a

polynomial

is again a polynomial, it is clear that

W

is

closed under

that multiplication.

The equivalence of these definitions can be seen by replacing the

generators

Q

i

with left multiplication by the indeterminates

X

i

,
the generators

P

i

with the partial differential operator

∂

i

, and the

tensor product

with operator multiplication, and
observing that

∂

i

⁡

X

j

-

X

j

⁢

∂

i

=

δ

i

⁢

j

. If, however,
the

characteristic

p

of

F

is positive, the resulting

homomorphism

to

W

is not

injective

, since for example the expressions

∂

i

p

and

X

i

n

commute, while

P

i

⊗

p

and

Q

i

⊗

n

do not.

Title

Weyl algebra

Canonical name

WeylAlgebra

Date of creation

2013-03-22 15:27:19

Last modified on

2013-03-22 15:27:19

Owner

GrafZahl (9234)

Last modified by

GrafZahl (9234)

Numerical id

5

Author

GrafZahl (9234)

Entry type

Definition

Classification

msc 16S36

Classification

msc 16S32

Related topic

HeisenbergAlgebra

Related topic

UniversalEnvelopingAlgebra

Generated on Fri Feb 9 18:42:55 2018 by

LaTeXML
generalized matrix ring

generalized matrix ring

Let

I

be an

indexing set

. A

ring of

I

×

I

generalized matrices

is a ring

R

with a decompostion (as an

additive group

)

R

=

⊕

i

,

j

∈

I

R

i

⁢

j

,

such that

R

i

⁢

j

⁢

R

k

⁢

l

⊆

R

i

⁢

l

if

j

=

k

and

R

i

⁢

j

⁢

R

k

⁢

l

=

0

if

j

≠

k

.

If

I

is finite, then we usually replace it by its

cardinal

n

and speak of a ring of

n

×

n

generalized matrices with

components

R

i

⁢

j

for

i

≤

i

,

j

≤

n

.

If we arrange the components

R

i

⁢

j

as follows:

(

R

11

R

12

…

R

1

⁢

n

R

21

R

22

…

R

2

⁢

n

⋮

⋮

⋱

⋮

R

n

⁢

1

R

n

⁢

2

…

R

n

⁢

n

)

and we write elements of

R

in the same fashion, then the

multiplication

in

R

follows
the same pattern as ordinary

matrix multiplication

.

Note that

R

i

⁢

j

is an

R

i

⁢

i

-

R

j

⁢

j

-

bimodule

,
and the multiplication of

elements

induces

homomorphisms

R

i

⁢

j

⊗

R

j

⁢

j

R

j

⁢

k

→

R

i

⁢

k

for all

i

,

j

,

k

.

Conversely

, given a

collection

of rings

R

i

,
and for each

i

≠

j

an

R

i

-

R

j

-bimodule

R

i

⁢

j

,
and for each

i

,

j

,

k

with

i

≠

j

and

j

≠

k

a homomorphism

R

i

⁢

j

⊗

R

j

R

j

⁢

k

→

R

i

⁢

k

,
we can construct a

generalized matrix ring

structure

on

R

=

⊕

i

,

j

R

i

⁢

j

,

where we take

R

i

⁢

i

=

R

i

.

Title

generalized matrix ring

Canonical name

GeneralizedMatrixRing

Date of creation

2013-03-22 14:39:14

Last modified on

2013-03-22 14:39:14

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

4

Author

mclase (549)

Entry type

Definition

Classification

msc 16S50

Generated on Fri Feb 9 18:42:58 2018 by

LaTeXML
complete set of primitive orthogonal idempotents

complete set of primitive orthogonal idempotents

Let

A

be a unital algebra over a field

k

. Recall that

e

∈

A

is an

idempotent

iff

e

2

=

e

. If

e

1

,

e

2

∈

A

are idempotents, then we will say that they are

orthogonal

iff

e

1

⁢

e

2

=

e

2

⁢

e

1

=

0

. Furthermore an idempotent

e

∈

A

is called

primitive

iff

e

cannot be written as a sum

e

=

e

1

+

e

2

where both

e

1

,

e

2

∈

A

are nonzero idempotents. An idempotent is called

trivial

iff it is either

0

or

1

.

Now assume that

A

is an

algebra

such that

A

=

M

1

⊕

M

2

as

right modules

and

1

=

m

1

+

m

2

for some

m

1

∈

M

1

,

m

2

∈

M

2

. Then

m

1

,

m

2

are orthogonal idempotents in

A

and

M

1

=

m

1

⁢

A

,

M

2

=

m

2

⁢

A

. Furthermore

M

i

is

indecomposable

(as a right module) if and only if

m

i

is primitive. This can be easily generalized to any number (but finite) of summands.

If

A

is additionally

finite-dimensional

, then

A

=

P

1

⊕

⋯

⊕

P

n

for some (unique up to

isomorphism

) right (ideals)

indecomposable modules

P

i

. It follows from the preceding that

P

i

=

e

i

⁢

A

for some

e

i

∈

A

and

{

e

1

,

…

,

e

n

}

is a set of pairwise orthogonal, primitive idempotents. This set is called

the

complete set of primitive orthogonal idempotents

of

A

.

Title

complete set of primitive orthogonal idempotents

Canonical name

CompleteSetOfPrimitiveOrthogonalIdempotents

Date of creation

2013-03-22 19:17:38

Last modified on

2013-03-22 19:17:38

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Definition

Classification

msc 16S99

Classification

msc 20C99

Classification

msc 13B99

Generated on Fri Feb 9 18:43:06 2018 by

LaTeXML
corner of a ring

corner of a ring

Does there exist a subset

S

of a ring

R

which is a ring with a

multiplicative identity

, but not a

subring

of

R

?

Let

R

be a ring without the

assumption

that

R

has a multiplicative identity. Further, assume that

e

is an

idempotent

of

R

. Then the subset of the form

e

⁢

R

⁢

e

is called a

corner

of the ring

R

.

It’s not hard to see that

e

⁢

R

⁢

e

is a ring with

e

as its multiplicative identity:

1.

e

⁢

a

⁢

e

+

e

⁢

b

⁢

e

=

e

⁢

(

a

+

b

)

⁢

e

∈

e

⁢

R

⁢

e

,

2.

0

=

e

⁢

0

⁢

e

∈

e

⁢

R

⁢

e

,

3.

e

⁢

(

-

a

)

⁢

e

is the

additive inverse

of

e

⁢

a

⁢

e

in

e

⁢

R

⁢

e

,

4.

(

e

⁢

a

⁢

e

)

⁢

(

e

⁢

b

⁢

e

)

=

e

⁢

(

a

⁢

e

⁢

b

)

⁢

e

∈

e

⁢

R

⁢

e

, and

5.

e

=

e

⁢

e

=

e

⁢

e

⁢

e

∈

e

⁢

R

⁢

e

, with

e

⁢

(

e

⁢

a

⁢

e

)

=

e

⁢

a

⁢

e

=

(

e

⁢

a

⁢

e

)

⁢

e

, for any

e

⁢

a

⁢

e

∈

e

⁢

R

⁢

e

.

If

R

has no multiplicative identity, then any corner of

R

is a

proper subset

of

R

which is a ring and not a subring of

R

. If

R

has 1 as its multiplicative identity and if

e

≠

1

is an idempotent, then the

e

⁢

R

⁢

e

is not a subring of

R

as they don’t share the same multiplicative identity. In this case, the corner

e

⁢

R

⁢

e

is said to be

proper

. If we set

f

=

1

-

e

, then

f

⁢

R

⁢

f

is also a proper corner of

R

.

Remark.

If

R

has 1 with

e

≠

1

an idempotent. Then corners

S

=

e

⁢

R

⁢

e

and

T

=

f

⁢

R

⁢

f

, where

f

=

1

-

e

, are

direct summands

(as modules over

ℤ

) of

R

via a

Peirce decomposition

.

References

1

I. Kaplansky,

Rings of

Operators

, W. A. Benjamin, Inc., New York, 1968.

Title

corner of a ring

Canonical name

CornerOfARing

Date of creation

2013-03-22 15:43:56

Last modified on

2013-03-22 15:43:56

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16S99

Related topic

UnityOfSubring

Generated on Fri Feb 9 18:43:08 2018 by

LaTeXML
finite dimensional modules over algebra

finite dimensional modules over algebra

Assume that

k

is a field,

A

is a

k

-

algebra

and

M

is a

A

-module over

k

. In particular

M

is a

A

-module and a

vector space

over

k

, thus we may speak about

M

being

finitely generated

as

A

-module and finite dimensional as a vector space. These two

concepts

are related as follows:

Proposition

.

Assume that

A

and

M

are both unital and additionaly

A

is finite dimensional. Then

M

is finite dimensional vector space if and only if

M

is finitely generated

A

-module.

Proof.

,,

⇒

” Of course if

M

is finite dimensional, then there exists basis

{

x

1

,

…

,

x

n

}

⊂

M

.

Thus every element of

M

can be (uniquely) expressed in the form

∑

i

=

1

n

λ

i

⋅

x

i

which is equal to

∑

i

=

1

n

(

λ

i

⋅

1

)

⋅

x

i

since

M

and

A

are unital. This

completes

this

implication

, because

λ

i

⋅

1

∈

A

for all

i

.

,,

⇐

” Assume that

M

is finitely generated

A

-module. In particular there is a subset

{

x

1

,

…

,

x

n

}

⊂

M

such that every element of

M

is of the form

∑

i

=

1

n

a

i

⋅

x

i

with all

a

i

∈

A

. Let

m

∈

M

be with the

decomposition

as above. Now

A

is finite dimensional, so there is a subset

{

y

1

,

…

,

y

t

}

⊂

A

which is a

k

-basis of

A

. In particular for each

i

we have

a

i

=

∑

j

=

1

t

λ

i

⁢

j

⋅

y

j

with

λ

i

⁢

j

∈

k

. Thus we obtain

m

=

∑

i

=

1

n

a

i

⋅

x

i

=

∑

i

=

1

n

(

∑

j

=

1

t

λ

i

⁢

j

⋅

y

j

)

⋅

x

i

=

=

∑

i

=

1

n

∑

j

=

1

t

λ

i

⁢

j

⋅

(

y

j

⋅

x

i

)

which shows, that all

y

j

⋅

x

i

∈

M

together make a set of

generators

of

M

over

k

(note that

y

j

and

x

i

are independent on

m

). Since it is finite, then

M

is finite dimensional and the proof is complete.

□

Title

finite dimensional modules over algebra

Canonical name

FiniteDimensionalModulesOverAlgebra

Date of creation

2013-03-22 19:16:35

Last modified on

2013-03-22 19:16:35

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Definition

Classification

msc 16S99

Classification

msc 20C99

Classification

msc 13B99

Generated on Fri Feb 9 18:43:10 2018 by

LaTeXML
functoriality of the Burnside ring

functoriality of the Burnside ring

We wish to show how the

Burnside ring

Ω

can be turned into a

contravariant functor

from the

category

of

finite groups

into the category of

commutative

,

unital rings

.

Let

G

and

H

be finite groups. We already know how

Ω

acts on objects of the category of finite groups. Assume that

f

:

G

→

H

is a

group homomorphism

. Furthermore let

X

be a

H

-set. Then

X

can be naturally equiped with a

G

-set

structure

via

function

:

(

g

,

x

)

⟼

f

⁢

(

g

)

⁢

x

.

The set

X

equiped with this

group action

will be denoted by

X

f

.

Therefore a group homomorphism

f

:

G

→

H

induces a

ring homomorphism

Ω

⁢

(

f

)

:

Ω

⁢

(

H

)

→

Ω

⁢

(

G

)

such that

Ω

⁢

(

f

)

⁢

(

[

X

]

-

[

Y

]

)

=

[

X

f

]

-

[

Y

f

]

.

One can easily check that this turns

Ω

into a contravariant functor.

Title

functoriality of the Burnside ring

Canonical name

FunctorialityOfTheBurnsideRing

Date of creation

2013-03-22 18:08:06

Last modified on

2013-03-22 18:08:06

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

5

Author

joking (16130)

Entry type

Derivation

Classification

msc 16S99

Generated on Fri Feb 9 18:43:12 2018 by

LaTeXML