aarjaneiro/planetmath huggingface dataset:
center (rings)

center (rings)

If

A

is a ring, the center of

A

, sometimes denoted

Z

⁡

(

A

)

, is the set of all elements in

A

that commute with all other elements of

A

. That is,

Z

⁡

(

A

)

=

{

a

∈

A

∣

a

⁢

x

=

x

⁢

a

⁢

⁢

∀

x

∈

A

}

Note that

0

∈

Z

⁡

(

A

)

so the center is non-empty. If we assume that

A

is a ring with a

multiplicative

unity

1

, then

1

is in the center as well. The center of

A

is also a subring of

A

.

Title

center (rings)

Canonical name

Centerrings

Date of creation

2013-03-22 12:45:29

Last modified on

2013-03-22 12:45:29

Owner

drini (3)

Last modified by

drini (3)

Numerical id

6

Author

drini (3)

Entry type

Definition

Classification

msc 16U70

Synonym

center

Related topic

GroupCentre

Generated on Fri Feb 9 18:43:52 2018 by

LaTeXML
anti-idempotent

anti-idempotent

An element

x

of a ring is called an

anti-idempotent

element, or simply an anti-idempotent if

x

2

=

-

x

.

The term is most often used in

linear algebra

. Every anti-idempotent matrix over a field is

diagonalizable

. Two anti-idempotent matrices are

similar

if and only if they have the same rank.

Title

anti-idempotent

Canonical name

Antiidempotent

Date of creation

2013-03-22 13:47:22

Last modified on

2013-03-22 13:47:22

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

4

Author

mathcam (2727)

Entry type

Definition

Classification

msc 16U99

Related topic

LinearInvolution

Generated on Fri Feb 9 18:43:56 2018 by

LaTeXML
behavior exists uniquely (finite case)

behavior exists uniquely (finite case)

The following is a proof that

behavior

exists uniquely for any finite

cyclic ring

R

.

Proof.

Let

n

be the order (

http://planetmath.org/OrderRing

) of

R

and

r

be a

generator

(

http://planetmath.org/Generator

) of the

additive group

of

R

. Then there exists

a

∈

ℤ

with

r

2

=

a

⁢

r

. Let

k

=

gcd

⁡

(

a

,

n

)

and

b

∈

ℤ

with

a

=

b

⁢

k

. Since

gcd

⁡

(

b

,

n

)

=

1

, there exists

c

∈

ℤ

with

b

⁢

c

≡

1

⁢

mod

⁡

n

. Since

gcd

⁡

(

c

,

n

)

=

1

,

c

⁢

r

is a generator of the additive group of

R

. Since

(

c

⁢

r

)

2

=

c

2

⁢

r

2

=

c

2

⁢

(

a

⁢

r

)

=

c

2

⁢

(

b

⁢

k

⁢

r

)

=

c

⁢

(

b

⁢

c

)

⁢

(

k

⁢

r

)

=

k

⁢

(

c

⁢

r

)

, it follows that

k

is a behavior of

R

. Thus, existence of behavior has been proven.

Let

g

and

h

be behaviors of

R

. Then there exist generators

s

and

t

of the additive group of

R

such that

s

2

=

g

⁢

s

and

t

2

=

h

⁢

t

. Since

t

is a generator of the additive group of

R

, there exists

w

∈

ℤ

with

gcd

⁡

(

w

,

n

)

=

1

such that

t

=

w

⁢

s

.

Note that

(

h

⁢

w

)

⁢

s

=

h

⁢

(

w

⁢

s

)

=

h

⁢

t

=

t

2

=

(

w

⁢

s

)

2

=

w

2

⁢

s

2

=

w

2

⁢

(

g

⁢

s

)

=

(

g

⁢

w

2

)

⁢

s

. Thus,

g

⁢

w

2

≡

h

⁢

w

⁢

mod

⁡

n

. Recall that

gcd

⁡

(

w

,

n

)

=

1

. Therefore,

g

⁢

w

≡

h

⁢

mod

⁡

n

. Since

g

and

h

are both positive

divisors

of

n

and

gcd

⁡

(

w

,

n

)

=

1

, it follows that

g

=

gcd

⁡

(

g

,

n

)

=

gcd

⁡

(

g

⁢

w

,

n

)

=

gcd

⁡

(

h

,

n

)

=

h

. Thus, uniqueness of behavior has been proven.
∎

Note that it has also been shown that, if

R

is a finite cyclic ring of order

n

,

r

is a generator of the additive group of

R

, and

a

∈

ℤ

with

r

2

=

a

⁢

r

, then the behavior of

R

is

gcd

⁡

(

a

,

n

)

.

Title

behavior exists uniquely (finite case)

Canonical name

BehaviorExistsUniquelyfiniteCase

Date of creation

2013-03-22 16:02:35

Last modified on

2013-03-22 16:02:35

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

13

Author

Wkbj79 (1863)

Entry type

Proof

Classification

msc 16U99

Classification

msc 13M05

Classification

msc 13A99

Generated on Fri Feb 9 18:43:59 2018 by

LaTeXML
central idempotent

central idempotent

Let

R

be a ring. An

element

e

∈

R

is called a

central idempotent

if it is an

idempotent

and is in the center

Z

⁢

(

R

)

of

R

.

It is well-known that if

e

∈

R

is an idempotent, then

e

⁢

R

⁢

e

has the

structure

of a

ring with unity

, with

e

being the unity. Thus, if

e

is

central

,

e

⁢

R

⁢

e

=

e

⁢

R

=

R

⁢

e

is a ring with unity

e

.

It is easy to see that the

operation

of ring multiplication preserves central idempotency: if

e

,

f

are central idempotents, so is

e

⁢

f

. In

addition

, if

R

has a multiplicative identity

1

, then

f

:=

1

-

e

is also a central idempotent. Furthermore, we may characterize central idempotency in a ring with

1

as follows:

Proposition 1

.

An idempotent

e

in a ring

R

with

1

is central iff

e

⁢

R

⁢

f

=

f

⁢

R

⁢

e

=

0

, where

f

=

1

-

e

.

Proof.

If

e

is central, then clearly

e

⁢

R

⁢

f

=

f

⁢

R

⁢

e

=

0

.

Conversely

, for any

r

∈

R

, we have

e

⁢

r

=

e

⁢

r

-

e

⁢

r

⁢

f

=

e

⁢

r

⁢

(

1

-

f

)

=

e

⁢

r

⁢

e

=

(

1

-

f

)

⁢

r

⁢

e

=

r

⁢

e

-

f

⁢

r

⁢

e

=

r

⁢

e

.
∎

Another interesting fact about central idempotents in a ring with unity is the following:

Proposition 2

.

The set

C

of all central idempotents of a ring

R

with

1

has the structure of a

Boolean ring

.

Proof.

First, note that

0

,

1

∈

C

. Next, for

e

,

f

∈

C

, we define addition

⊕

and multiplication

⊙

on

C

as follows:

e

⊕

f

:=

e

+

f

-

e

⁢

f

and

e

⊙

f

:=

e

⁢

f

.

As discussed above,

⊕

and

⊙

are well-defined (as

C

is

closed under

these operations). In addition, for any

e

,

f

,

g

∈

C

, we have

1.

(

C

,

1

,

⊙

)

is a

commutative monoid

, in which every element is an idempotent (with respect to

⊙

). This fact is clear.

2.

⊕

is

commutative

, since

C

⊆

Z

⁢

(

R

)

.

3.

⊕

is

associative

:

e

⊕

(

f

⊕

g

)

=

e

+

(

f

+

g

-

f

⁢

g

)

-

e

⁢

(

f

+

g

-

f

⁢

g

)

=

e

+

f

+

g

-

e

⁢

f

-

f

⁢

g

-

e

⁢

g

+

e

⁢

f

⁢

g

=

(

e

+

f

-

e

⁢

f

)

+

g

-

(

e

+

f

-

e

⁢

f

)

⁢

g

=

(

e

⊕

f

)

⊕

g

.

4.

⊙

distributes over

⊕

: we only need to show left distributivity (since

⊙

is commutative by

1

above):

e

⊙

(

f

⊕

g

)

=

e

⁢

(

f

+

g

-

f

⁢

g

)

=

e

⁢

f

+

e

⁢

g

-

e

⁢

f

⁢

g

=

e

⁢

f

+

e

⁢

g

-

e

⁢

e

⁢

f

⁢

g

=

e

⁢

f

+

e

⁢

g

-

e

⁢

f

⁢

e

⁢

g

=

e

⁢

f

⊕

e

⁢

g

=

(

e

⊙

f

)

⊕

(

e

⊙

g

)

.

This shows that

(

C

,

0

,

1

,

⊕

,

⊙

)

is a Boolean ring.
∎

Title

central idempotent

Canonical name

CentralIdempotent

Date of creation

2013-03-22 19:13:07

Last modified on

2013-03-22 19:13:07

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16U99

Classification

msc 20M99

Generated on Fri Feb 9 18:44:08 2018 by

LaTeXML
criterion for cyclic rings to be principal ideal rings

criterion for cyclic rings to be principal ideal rings

Theorem

.

A

cyclic ring

is a

principal ideal ring

if and only if it has a

multiplicative identity

.

Proof.

Let

R

be a cyclic ring. If

R

has a multiplicative identity

u

, then

u

generates

(

http://planetmath.org/

Generator

) the

additive group

of

R

. Let

I

be an ideal of

R

. Since

{

0

R

}

is principal, it may be assumed that

I

contains a nonzero

element

. Let

n

be the smallest

natural number

such that

n

⁢

u

∈

I

. The

inclusion

⟨

n

⁢

u

⟩

⊆

I

is

trivial

. Let

t

∈

I

. Since

t

∈

R

, there exists

a

∈

ℤ

with

t

=

a

⁢

u

. By the

division algorithm

, there exists

q

,

r

∈

ℤ

with

0

≤

r

<

n

such that

a

=

q

⁢

n

+

r

. Thus,

t

=

a

⁢

u

=

(

q

⁢

n

+

r

)

⁢

u

=

(

q

⁢

n

)

⁢

u

+

r

⁢

u

=

q

⁢

(

n

⁢

u

)

+

r

⁢

u

. Since

r

⁢

u

=

t

-

q

⁢

(

n

⁢

u

)

∈

I

, by choice of

n

, it must be the case that

r

=

0

. Thus,

t

=

q

⁢

(

n

⁢

u

)

. Hence,

⟨

n

⁢

u

⟩

=

I

, and

R

is a principal ideal ring.

Conversely

, if

R

is a principal ideal ring, then

R

is a

principal ideal

. Let

k

be the

behavior

of

R

and

r

be a generator (

http://planetmath.org/Generator

) of the additive group of

R

such that

r

2

=

k

⁢

r

. Since

R

is principal, there exists

s

∈

R

such that

⟨

s

⟩

=

R

. Let

a

∈

ℤ

such that

s

=

a

⁢

r

. Since

r

∈

R

=

⟨

s

⟩

, there exists

t

∈

R

with

s

⁢

t

=

r

. Let

b

∈

ℤ

such that

t

=

b

⁢

r

. Then

r

=

s

⁢

t

=

(

a

⁢

r

)

⁢

(

b

⁢

r

)

=

(

a

⁢

b

)

⁢

r

2

=

(

a

⁢

b

)

⁢

(

k

⁢

r

)

=

(

a

⁢

b

⁢

k

)

⁢

r

. If

R

is

infinite

, then

a

⁢

b

⁢

k

=

1

, in which case

k

=

1

since

k

is nonnegative. If

R

is finite, then

a

⁢

b

⁢

k

≡

1

⁢

mod

⁡

|

R

|

, in which case

k

=

1

since

k

is a

positive

divisor

of

|

R

|

. In either case,

R

has behavior one, and it follows that

R

has a multiplicative identity.
∎

Title

criterion for cyclic rings to be principal ideal rings

Canonical name

CriterionForCyclicRingsToBePrincipalIdealRings

Date of creation

2013-03-22 15:57:03

Last modified on

2013-03-22 15:57:03

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

15

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 16U99

Classification

msc 13A99

Classification

msc 13F10

Related topic

CyclicRing3

Related topic

PrincipalIdealRing

Related topic

MultiplicativeIdentityOfACyclicRingMustBeAGenerator

Related topic

CyclicRingsOfBehaviorOne

Generated on Fri Feb 9 18:44:11 2018 by

LaTeXML
cyclic rings that are isomorphic to k⁢ℤ_{k⁢n}

cyclic rings that are isomorphic to

k

⁢

ℤ

k

⁢

n

Corollary

.

A finite

cyclic ring

of order (

http://planetmath.org/OrderRing

)

n

with

behavior

k

is

isomorphic

to

k

⁢

Z

k

⁢

n

.

Proof.

Note that

k

⁢

ℤ

k

⁢

n

is a cyclic ring and that

k

is a

generator

of its

additive group

. As groups,

k

⁢

ℤ

k

⁢

n

and

ℤ

n

are isomorphic. Thus,

k

⁢

ℤ

k

⁢

n

has order

n

. Since

k

2

=

k

⁢

(

k

)

, then

k

⁢

ℤ

has behavior

k

.
∎

Title

cyclic rings that are isomorphic to

k

⁢

ℤ

k

⁢

n

Canonical name

CyclicRingsThatAreIsomorphicToKmathbbZkn

Date of creation

2013-03-22 16:02:45

Last modified on

2013-03-22 16:02:45

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

10

Author

Wkbj79 (1863)

Entry type

Corollary

Classification

msc 16U99

Classification

msc 13M05

Classification

msc 13A99

Related topic

MathbbZ_n

Generated on Fri Feb 9 18:44:13 2018 by

LaTeXML
Dedekind-finite

Dedekind-finite

A ring

R

is

Dedekind-finite

if for

a

,

b

∈

R

, whenever

a

⁢

b

=

1

implies

b

⁢

a

=

1

.

Of course, every

commutative ring

is Dedekind-finite. Therefore, the theory of Dedekind finiteness is

trivial

in this case. Some other examples are

1.

any

ring of endomorphisms

over a finite dimensional

vector space

(over a field)

2.

any

division ring

3.

any ring of matrices over a division ring

4.

finite

direct product

of Dedekind-finite rings

5.

by the last three examples, any

semi-simple ring

is Dedekind-finite.

6.

any ring

R

with the property that there is a

natural number

n

such that

x

n

=

0

for every

nilpotent element

x

∈

R

The finite dimensionality in the first example can not be extended to the

infinite

case. Lam in

[

1

]

gave an example of a ring that is not Dedekind-finite arising out of the ring of endomorphisms over an infinite dimensional vector space (over a field).

References

1

T. Y. Lam,

A First Course in

Noncommutative Rings

, Springer-Verlag, New York (1991).

2

T. Y. Lam,

Lectures on Modules and Rings

, Springer-Verlag, New York (1999).

Title

Dedekind-finite

Canonical name

Dedekindfinite

Date of creation

2013-03-22 14:18:23

Last modified on

2013-03-22 14:18:23

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16U99

Synonym

von Neumann-finite

Generated on Fri Feb 9 18:44:15 2018 by

LaTeXML
idempotent

idempotent

An element

x

of a ring is called an

idempotent element

, or simply an

idempotent

if

x

2

=

x

.

The set of idempotents of a ring can be partially ordered by putting

e

≤

f

iff

e

=

e

⁢

f

=

f

⁢

e

.

The element

0

is a minimum element in this

partial order

. If the ring has an

identity element

,

1

, then

1

is a maximum element in this partial order.

Since the above definitions refer only to the multiplicative structure of the ring, they also hold for

semigroups

(with the proviso, of course, that a semigroup may have neither a

zero element

nor an identity element). In the special case of a

semilattice

, this partial order is the same as the one described in the entry for semilattice.

If a ring has an

identity

, then

1

-

e

is always an idempotent whenever

e

is an idempotent, and

e

⁢

(

1

-

e

)

=

(

1

-

e

)

⁢

e

=

0

.

In a ring with an identity, two idempotents

e

and

f

are called a

pair of orthogonal idempotents

if

e

+

f

=

1

, and

e

⁢

f

=

f

⁢

e

=

0

. Obviously, this is just a fancy way of saying that

f

=

1

-

e

.

More generally, a set

{

e

1

,

e

2

,

…

,

e

n

}

of idempotents is called a

complete set of orthogonal idempotents

if

e

i

⁢

e

j

=

e

j

⁢

e

i

=

0

whenever

i

≠

j

and if

1

=

e

1

+

e

2

+

…

+

e

n

.

If

{

e

1

,

e

2

,

…

,

e

n

}

is a complete set of orthogonal idempotents,
and in addition each

e

i

is in the centre of

R

, then each

R

⁢

e

i

is a

subring

, and

R

≅

R

⁢

e

1

×

R

⁢

e

2

×

…

×

R

⁢

e

n

.

Conversely, whenever

R

1

×

R

2

×

…

×

R

n

is a

direct
product

of

rings with identities

, write

e

i

for the element of the

product

corresponding to the identity element of

R

i

. Then

{

e

1

,

e

2

,

…

,

e

n

}

is a complete set of central orthogonal idempotents of the product ring.

When a complete set of orthogonal idempotents is not central, there is a more complicated : see the entry on the

Peirce decomposition

for the details.

Title

idempotent

Canonical name

Idempotent

Date of creation

2013-03-22 13:07:27

Last modified on

2013-03-22 13:07:27

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

11

Author

mclase (549)

Entry type

Definition

Classification

msc 16U99

Classification

msc 20M99

Synonym

idempotent element

Related topic

Semilattice

Related topic

Idempotency

Defines

orthogonal idempotents

Defines

complete set of orthogonal idempotents

Generated on Fri Feb 9 18:44:17 2018 by

LaTeXML
prime ring

prime ring

A ring

R

is said to be a

prime ring

if the

zero ideal

is a

prime ideal

.

If a prime ring

R

is

commutative

, then it is a

cancellation ring

. If in

R

has a

multiplicative identity

1

≠

0

, then it is an

integral domain

.

Title

prime ring

Canonical name

PrimeRing

Date of creation

2013-03-22 11:51:05

Last modified on

2013-03-22 11:51:05

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

10

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 16U99

Classification

msc 16N60

Related topic

ZeroIdeal

Generated on Fri Feb 9 18:44:31 2018 by

LaTeXML
the multiplicative identity of a cyclic ring must be a generator

the multiplicative identity of a cyclic ring must be a generator

Theorem

.

Let

R

be a

cyclic ring

with

multiplicative identity

u

. Then

u

generates

(

http://planetmath.org/

Generator

) the

additive group

of

R

.

Proof.

Let

k

be the

behavior

of

R

. Then there exists a generator (

http://planetmath.org/Generator

)

r

of the additive group of

R

such that

r

2

=

k

⁢

r

. Let

a

∈

ℤ

with

u

=

a

⁢

r

. Then

r

=

u

⁢

r

=

(

a

⁢

r

)

⁢

r

=

a

⁢

r

2

=

a

⁢

(

k

⁢

r

)

=

(

a

⁢

k

)

⁢

r

. If

R

is

infinite

, then

a

⁢

k

=

1

, causing

a

=

k

=

1

since

k

is a nonnegative integer. If

R

is finite, then

a

⁢

k

≡

1

⁢

mod

⁡

|

R

|

. Thus,

gcd

⁡

(

k

,

|

R

|

)

=

1

. Since

k

divides

|

R

|

,

k

=

1

. Therefore,

a

≡

1

⁢

mod

⁡

|

R

|

. In either case,

u

=

r

.
∎

Note that it was also proven that, if a cyclic ring has a multiplicative identity, then it has behavior one. Its converse is also true. See this theorem (

http://planetmath.org/CyclicRingsOfBehaviorOne

) for more details.

Title

the multiplicative identity of a cyclic ring must be a generator

Canonical name

TheMultiplicativeIdentityOfACyclicRingMustBeAGenerator

Date of creation

2013-03-22 15:56:59

Last modified on

2013-03-22 15:56:59

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

16

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 16U99

Classification

msc 13F10

Classification

msc 13A99

Related topic

CyclicRing3

Related topic

CriterionForCyclicRingsToBePrincipalIdealRings

Related topic

CyclicRingsOfBehaviorOne

Generated on Fri Feb 9 18:44:34 2018 by

LaTeXML
zero ring

zero ring

A ring is a

zero ring

if the

product

of any two elements is the

additive identity

(or zero).

Zero rings are

commutative

under

multiplication

. For if

Z

is a zero ring,

0

Z

is its additive identity, and

x

,

y

∈

Z

, then

x

⁢

y

=

0

Z

=

y

⁢

x

.

Every zero ring is a

nilpotent ring

. For if

Z

is a zero ring, then

Z

2

=

{

0

Z

}

.

Since every

subring

of a ring must contain its

zero element

, every subring of a ring is an ideal, and a zero ring has no

prime ideals

.

The simplest zero ring is

ℤ

1

=

{

0

}

. Up to

isomorphism

(

http://planetmath.org/RingIsomorphism

), this is the only zero ring that has a

multiplicative identity

.

Zero rings exist in . They can be constructed from any ring. If

R

is a ring, then

{

(

r

-

r

r

-

r

)

|

r

∈

R

}

considered as a subring of

𝐌

2

⁢

x

⁡

2

⁢

(

R

)

(with standard

matrix addition

and multiplication) is a zero ring. Moreover, the cardinality of this subset of

𝐌

2

⁢

x

⁡

2

⁢

(

R

)

is the same as that of

R

.

Moreover, zero rings can be constructed from any

abelian group

. If

G

is a

group with

identity

e

G

, it can be made into a zero ring by declaring its

addition

to be its

group operation

and defining its multiplication by

a

⋅

b

=

e

G

for any

a

,

b

∈

G

.

Every finite zero ring can be written as a

direct product

of

cyclic rings

, which must also be zero rings themselves. This follows from the fundamental theorem of finite abelian groups (

http://planetmath.org/FundamentalTheoremOfFinitelyGeneratedAbelianGroups

). Thus, if

p

1

,

…

,

p

m

are distinct primes,

a

1

,

…

,

a

m

are

positive integers

, and

n

=

∏

j

=

1

m

p

j

a

j

, then the number of zero rings of order (

http://planetmath.org/Order

)

n

is

∏

j

=

1

m

p

⁢

(

a

j

)

, where

p

denotes the

partition function

(

http://planetmath.org/PartitionFunction2

).

Title

zero ring

Canonical name

ZeroRing

Date of creation

2013-03-22 13:30:19

Last modified on

2013-03-22 13:30:19

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

26

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 16U99

Classification

msc 13M05

Classification

msc 13A99

Related topic

ZeroVectorSpace

Related topic

Unity

Generated on Fri Feb 9 18:44:36 2018 by

LaTeXML
faithful group action

faithful group action

Let

A

be a

G

-set, that is, a set acted upon by a group

G

with action

ψ

:

G

×

A

→

A

. Then for any

g

∈

G

, the map

m

g

:

A

→

A

defined by

m

g

⁢

(

x

)

=

ψ

⁢

(

g

,

x

)

is a

permutation

of

A

(in other words, a

bijective function

from

A

to itself) and so an element of

S

A

.
We can even get an

homomorphism

from

G

to

S

A

by the rule

g

↦

m

g

.

If for any pair

g

,

h

∈

G

g

≠

h

we have

m

g

≠

m

h

, in other words, the homomorphism

g

→

m

g

being

injective

, we say that the action is

faithful

.

Title

faithful group action

Canonical

name

FaithfulGroupAction

Date of creation

2013-03-22 14:02:23

Last modified on

2013-03-22 14:02:23

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 16W22

Classification

msc 20M30

Generated on Fri Feb 9 18:44:40 2018 by

LaTeXML
group action

group action

Let

G

be a group and let

X

be a set. A left

group action

is a

function

⋅

:

G

×

X

⟶

X

such that:

1.

1

G

⋅

x

=

x

for all

x

∈

X

2.

(

g

1

⁢

g

2

)

⋅

x

=

g

1

⋅

(

g

2

⋅

x

)

for all

g

1

,

g

2

∈

G

and

x

∈

X

A right

group action

is a function

⋅

:

X

×

G

⟶

X

such that:

1.

x

⋅

1

G

=

x

for all

x

∈

X

2.

x

⋅

(

g

1

⁢

g

2

)

=

(

x

⋅

g

1

)

⋅

g

2

for all

g

1

,

g

2

∈

G

and

x

∈

X

There is a correspondence between left actions and right actions, given by associating the right action

x

⋅

g

with the left action

g

⋅

x

:=

x

⋅

g

-

1

. In many (but not all)

contexts

, it is useful to identify right actions with their corresponding left actions, and speak only of left actions.

Special types of group actions

A left action is said to be

effective

, or

faithful

, if the function

x

↦

g

⋅

x

is the

identity function

on

X

only when

g

=

1

G

.

A left action is said to be

transitive

if, for every

x

1

,

x

2

∈

X

, there exists a group

element

g

∈

G

such that

g

⋅

x

1

=

x

2

.

A left action is

free

if, for every

x

∈

X

, the only element of

G

that stabilizes

x

is the

identity

; that is,

g

⋅

x

=

x

implies

g

=

1

G

.

Faithful, transitive, and free right actions are defined similarly.

Title

group action

Canonical

name

GroupAction

Date of creation

2013-03-22 12:12:17

Last modified on

2013-03-22 12:12:17

Owner

djao (24)

Last modified by

djao (24)

Numerical id

10

Author

djao (24)

Entry type

Definition

Classification

msc 16W22

Classification

msc 20M30

Related topic

Group

Defines

effective

Defines

effective group action

Defines

faithful

Defines

faithful group action

Defines

transitive

Defines

transitive group action

Defines

left action

Defines

right action

Defines

faithfully

Defines

action

Defines

act on

Defines

acts on

Generated on Fri Feb 9 18:44:41 2018 by

LaTeXML
almost cocommutative bialgebra

almost cocommutative bialgebra

A

bialgebra

A

is called

almost cocommutative

if there is an unit

ℛ

∈

A

⊗

A

such that

ℛ

⁢

Δ

⁢

(

a

)

=

Δ

o

⁢

p

⁢

(

a

)

⁢

ℛ

where

Δ

o

⁢

p

is the opposite comultiplication (the usual comultiplication, composed with the flip map of the

tensor product

A

⊗

A

). The element

ℛ

is often called the

ℛ

-matrix of

A

.

The significance of the almost cocommutative condition is that

σ

V

,

W

=

σ

∘

ℛ

:

V

⊗

W

→

W

⊗

V

gives a

natural isomorphism

of bialgebra

representations

, where

V

and

W

are

A

-modules, making the

category

of

A

-modules into a quasi-tensor or braided monoidal category. Note that

σ

W

,

V

∘

σ

V

,

W

is not necessarily the

identity

(this is the

braiding

of the category).

Title

almost cocommutative bialgebra

Canonical name

AlmostCocommutativeBialgebra

Date of creation

2013-03-22 13:31:50

Last modified on

2013-03-22 13:31:50

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

5

Author

bwebste (988)

Entry type

Definition

Classification

msc 16W30

Generated on Fri Feb 9 18:44:43 2018 by

LaTeXML