aarjaneiro/planetmath huggingface dataset:
proof of properties of trace of a matrix

proof of properties of trace of a matrix

Proof of

Properties

:

1.

Let us check linearity. For sums we have

trace

⁡

(

A

+

B

)

=

∑

i

=

1

n

(

a

i

,

i

+

b

i

,

i

)

(property of matrix addition)

=

∑

i

=

1

n

a

i

,

i

+

∑

i

=

1

n

b

i

,

i

⁢

(property of sums)

=

trace

⁡

(

A

)

+

trace

⁡

(

B

)

.

Similarly,

trace

⁡

(

c

⁢

A

)

=

∑

i

=

1

n

c

⋅

a

i

,

i

⁢

(property of matrix scalar multiplication)

=

c

⋅

∑

i

=

1

n

a

i

,

i

⁢

(property of sums)

=

c

⋅

trace

⁡

(

A

)

.

2.

The second property follows since the

transpose

does not alter
the entries on the main diagonal.

3.

The proof of the third property follows by exchanging the

summation

order. Suppose

A

is a

n

×

m

matrix and

B

is a

m

×

n

matrix.
Then

trace

⁡

A

⁢

B

=

∑

i

=

1

n

∑

j

=

1

m

A

i

,

j

⁢

B

j

,

i

=

∑

j

=

1

m

∑

i

=

1

n

B

j

,

i

⁢

A

i

,

j

⁢

(changing summation order)

=

trace

⁡

B

⁢

A

.

4.

The last property is a consequence of Property 3 and the fact that

matrix multiplication

is
associative;

trace

⁡

(

B

-

1

⁢

A

⁢

B

)

=

trace

⁡

(

(

B

-

1

⁢

A

)

⁢

B

)

=

trace

⁡

(

B

⁢

(

B

-

1

⁢

A

)

)

=

trace

⁡

(

(

B

⁢

B

-

1

)

⁢

A

)

=

trace

⁡

(

A

)

.

Title

proof of properties of trace of a matrix

Canonical

name

ProofOfPropertiesOfTraceOfAMatrix

Date of creation

2013-03-22 13:42:54

Last modified on

2013-03-22 13:42:54

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

4

Author

Daume (40)

Entry type

Proof

Classification

msc 15A99

Generated on Fri Feb 9 18:34:19 2018 by

LaTeXML
Q is the prime subfield of any field of characteristic 0, proof that

Q is the prime subfield of any field of characteristic 0, proof that

The following two

propositions

show that

ℚ

can be embedded in any field of

characteristic

0

, while

𝔽

p

can be embedded in any field of characteristic

p

.

Proposition.

ℚ

is the

prime subfield

of any field of characteristic 0.

Proof.

Let

F

be a field of characteristic

0

.  We want to find a

one-to-one

field homomorphism

ϕ

:

ℚ

→

F

.  For

m

n

∈

ℚ

with

m

,

n

coprime

, define the

mapping

ϕ

that takes

m

n

into

m

⁢

1

F

n

⁢

1

F

∈

F

.  It is easy to check that

ϕ

is a well-defined

function

.  Furthermore, it is

elementary

to show

1.

additive

: for

p

,

q

∈

ℚ

,

ϕ

⁢

(

p

+

q

)

=

ϕ

⁢

(

p

)

+

ϕ

⁢

(

q

)

;

2.

multiplicative

: for

p

,

q

∈

ℚ

,

ϕ

⁢

(

p

⁢

q

)

=

ϕ

⁢

(

p

)

⁢

ϕ

⁢

(

q

)

;

3.

ϕ

⁢

(

1

)

=

1

F

, and

4.

ϕ

⁢

(

0

)

=

0

F

.

This shows that

ϕ

is a field homomorphism. Finally, if

ϕ

⁢

(

p

)

=

0

and

p

≠

0

, then

1

=

ϕ

⁢

(

1

)

=

ϕ

⁢

(

p

⁢

p

-

1

)

=

ϕ

⁢

(

p

)

⁢

ϕ

⁢

(

p

-

1

)

=

0

⋅

ϕ

⁢

(

p

-

1

)

=

0

, a

contradiction

.
∎

Proposition.

𝔽

p

(

≅

ℤ

/

p

⁢

ℤ

) is the prime subfield of any field of characteristic

p

.

Proof.

Let

F

be a field of characteristic

p

. The idea again is to find an injective field homomorphism, this time, from

𝔽

p

into

F

. Take

ϕ

to be the function that maps

m

∈

𝔽

p

to

m

⋅

1

F

. It is well-defined, for if

m

=

n

in

𝔽

p

, then

p

∣

(

m

-

n

)

, meaning

(

m

-

n

)

⁢

1

F

=

0

, or that

m

⋅

1

F

=

n

⋅

1

F

, (showing that one

element

in

𝔽

p

does not get “mapped” to more than one element in

F

). Since the above

argument

is reversible, we see that

ϕ

is one-to-one.

To

complete

the proof, we next show that

ϕ

is a field homomorphism. That

ϕ

⁢

(

1

)

=

1

F

and

ϕ

⁢

(

0

)

=

0

F

are clear from the definition of

ϕ

. Additivity and multiplicativity of

ϕ

are readily verified, as follows:

•

ϕ

⁢

(

m

+

n

)

=

(

m

+

n

)

⋅

1

F

=

m

⋅

1

F

+

n

⋅

1

F

=

ϕ

⁢

(

m

)

+

ϕ

⁢

(

n

)

;

•

ϕ

⁢

(

m

⁢

n

)

=

m

⁢

n

⋅

1

F

=

m

⁢

n

⋅

1

F

⋅

1

F

=

(

m

⋅

1

F

)

⁢

(

n

⋅

1

F

)

=

ϕ

⁢

(

m

)

⁢

ϕ

⁢

(

n

)

.

This shows that

ϕ

is a field homomorphism.
∎

Title

Q is the prime subfield of any field of characteristic 0, proof that

Canonical name

QIsThePrimeSubfieldOfAnyFieldOfCharacteristic0ProofThat

Date of creation

2013-03-22 15:39:57

Last modified on

2013-03-22 15:39:57

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

16

Author

CWoo (3771)

Entry type

Proof

Classification

msc 15A99

Classification

msc 12F99

Classification

msc 12E99

Classification

msc 12E20

Related topic

RationalNumbersAreRealNumbers

Defines

prime field

Generated on Fri Feb 9 18:34:22 2018 by

LaTeXML
quasipositive matrix

quasipositive matrix

A

square matrix

A

=

(

a

i

⁢

j

)

is a

quasipositive

matrix if it is nonnegative except perhaps on its main diagonal, i.e.,

a

i

⁢

j

≥

0

for

i

≠

j

.

Title

quasipositive matrix

Canonical name

QuasipositiveMatrix

Date of creation

2013-03-22 13:18:23

Last modified on

2013-03-22 13:18:23

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

6

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 15A99

Synonym

essentially nonnegative matrix

Generated on Fri Feb 9 18:34:24 2018 by

LaTeXML
spanning sets of dual space

spanning sets of dual space

Theorem

.

Let

X

be a

vector space

and

ϕ

1

,

…

,

ϕ

n

∈

X

*

be

functionals

belonging to the

dual space

.
A

linear functional

f

∈

X

*

belongs to the

linear span

of

ϕ

1

,

…

,

ϕ

n

if and only if

ker

⁡

f

⊇

⋂

i

=

1

n

ker

⁡

ϕ

i

.

ker

refers to the kernel.
Note that the domain

X

need not be

finite-dimensional

.

Proof.

The “only if” part is easy: if

f

=

∑

i

=

1

n

λ

i

⁢

ϕ

i

for some scalars

λ

i

, and

x

∈

X

is such that

ϕ

i

⁢

(

x

)

=

0

for all

i

, then clearly

f

⁢

(

x

)

=

0

too.

The “if” part will be proved by

induction

on

n

.

Suppose

ker

⁡

f

⊇

ker

⁡

ϕ

1

.
If

f

=

0

, then the result is

trivial

.
Otherwise, there exists

y

∈

X

such that

f

⁢

(

y

)

≠

0

.
By

hypothesis

, we also have

ϕ

1

⁢

(

y

)

≠

0

.
Every

z

∈

X

can be decomposed into

z

=

x

+

t

⁢

y

where

x

∈

ker

⁡

ϕ

1

⊆

ker

⁡

f

, and

t

is a scalar.
Indeed, just set

t

=

ϕ

1

⁢

(

z

)

/

ϕ

1

⁢

(

y

)

, and

x

=

z

-

t

⁢

y

.
Then we propose that

f

⁢

(

z

)

=

f

⁢

(

y

)

ϕ

1

⁢

(

y

)

⁢

ϕ

1

⁢

(

z

)

,

for all

z

∈

X

.

To check this

equation

, simply evaluate both sides using the

decomposition

z

=

x

+

t

⁢

y

.

Now suppose we have

ker

⁡

f

⊇

⋂

i

=

1

n

ker

⁡

ϕ

i

for

n

>

1

.
Restrict each of the functionals
to the

subspace

W

=

ker

⁡

ϕ

n

, so that

ker

⁡

f

|

W

⊇

⋂

i

=

1

n

-

1

ker

⁡

ϕ

i

|

W

.
By the

induction hypothesis

, there exist scalars

λ

1

,

…

,

λ

n

-

1

such that

f

|

W

=

∑

i

=

1

n

-

1

λ

i

⁢

ϕ

i

|

W

.
Then

ker

⁡

(

f

-

∑

i

=

1

n

-

1

λ

i

⁢

ϕ

i

)

⊇

W

=

ker

⁡

ϕ

n

, and the

argument

for the case

n

=

1

can be applied anew, to obtain the final

λ

n

.
∎

Title

spanning sets of dual space

Canonical

name

SpanningSetsOfDualSpace

Date of creation

2013-03-22 17:17:28

Last modified on

2013-03-22 17:17:28

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

6

Author

stevecheng (10074)

Entry type

Theorem

Classification

msc 15A99

Generated on Fri Feb 9 18:34:28 2018 by

LaTeXML
trace of a matrix

trace of a matrix

Definition

Let

A

=

(

a

i

,

j

)

be a

square matrix

of
order

n

.
The trace of the matrix is the sum of the main diagonal:

trace

⁡

(

A

)

=

∑

i

=

1

n

a

i

,

i

Notation:

The trace of a matrix

A

is also commonly denoted as

Tr

⁡

(

A

)

or

Tr

⁡

A

.

Properties:

1.

The trace is a

linear transformation

from the space of square matrices to
the

real numbers

. In other words, if

A

and

B

are square matrices with real (or

complex

) entries,
of same order and

c

is a scalar, then

trace

⁡

(

A

+

B

)

=

trace

⁡

(

A

)

+

trace

⁡

(

B

)

,

trace

⁡

(

c

⁢

A

)

=

c

⋅

trace

⁡

(

A

)

.

2.

For the

transpose

and

conjugate transpose

, we have for any
square matrix

A

with real (or complex) entries,

trace

⁡

(

A

t

)

=

trace

⁡

(

A

)

,

trace

⁡

(

A

∗

)

=

trace

⁡

(

A

)

¯

.

3.

If

A

and

B

are matrices such that

A

⁢

B

is a square matrix, then

trace

⁡

(

A

⁢

B

)

=

trace

⁡

(

B

⁢

A

)

.

For this reason it is possible to define the trace of a linear transformation, as the choice of basis does not affect the trace.
Thus, if

A

,

B

,

C

are matrices such that

A

⁢

B

⁢

C

is a square matrix, then

trace

⁡

(

A

⁢

B

⁢

C

)

=

trace

⁡

(

C

⁢

A

⁢

B

)

=

trace

⁡

(

B

⁢

C

⁢

A

)

.

4.

If

B

is in

invertible

square matrix of same order as

A

, then

trace

⁡

(

A

)

=

trace

⁡

(

B

-

1

⁢

A

⁢

B

)

.

In other words, the trace of

similar matrices

are equal.

5.

Let

A

be a square matrix of order

n

with real (or complex)
entries

a

i

⁢

j

. Then

trace

⁡

A

∗

⁢

A

=

trace

⁡

A

⁢

A

∗

=

∑

i

,

j

=

1

n

|

a

i

⁢

j

|

2

.

Here

∗

is the

complex conjugate

, and

|

⋅

|

is the

complex modulus

.
In particular,

trace

⁡

A

∗

⁢

A

≥

0

with equality if and only if

A

=

0

.
(See the

Frobenius matrix norm

.)

6.

Various

inequalities

for

trace

are given in

[

2

]

.

See the proof of properties of trace of a matrix.

References

1

The Trace of a Square Matrix. Paul Ehrlich, [online]

http://www.math.ufl.edu/ ehrlich/trace.html

http://www.math.ufl.edu/ ehrlich/trace.html

2

Z.P. Yang, X.X. Feng,

A note on the trace inequality for

products

of

Hermitian matrix

power

,
Journal of Inequalities in Pure and Applied Mathematics,
Volume 3, Issue 5, 2002, Article 78,

http://www.emis.de/journals/JIPAM/v3n5/082_02.html

online.

Title

trace of a matrix

Canonical name

TraceOfAMatrix

Date of creation

2013-03-22 11:59:56

Last modified on

2013-03-22 11:59:56

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

20

Author

Daume (40)

Entry type

Definition

Classification

msc 15A99

Related topic

ShursInequality

Generated on Fri Feb 9 18:34:31 2018 by

LaTeXML
additive inverse of one element times another element is the additive inverse of their product

additive inverse of one element times another element is the additive inverse of their product

Let

R

be a ring. For all

x

,

y

∈

R

(

-

x

)

⋅

y

=

x

⋅

(

-

y

)

=

-

(

x

⋅

y

)

All we need to prove is that

(

-

x

)

⋅

y

+

x

⋅

y

=

x

⋅

(

-

y

)

+

x

⋅

y

=

0

Now:

(

-

x

)

⋅

y

+

x

⋅

y

=

(

(

-

x

)

+

x

)

⋅

y

by

distributivity

.

Since

(

-

x

)

+

x

=

0

by definition and for all

y

,

0

⋅

y

=

0

we get:

(

-

x

)

⋅

y

+

x

⋅

y

=

0

⋅

y

=

0

and thus

(

-

x

)

⋅

y

=

-

(

x

⋅

y

)

For

x

⋅

(

-

y

)

, use the previous properties of rings to show that

x

⋅

(

-

y

)

+

x

⋅

y

=

x

⋅

(

(

-

y

)

+

y

)

=

x

⋅

0

=

0

and thus

x

⋅

(

-

y

)

=

-

(

x

⋅

y

)

Title

additive inverse

of one element times another element is the additive inverse of their product

Canonical name

AdditiveInverseOfOneElementTimesAnotherElementIsTheAdditiveInverseOfTheirProduct

Date of creation

2013-03-22 15:43:40

Last modified on

2013-03-22 15:43:40

Owner

cvalente (11260)

Last modified by

cvalente (11260)

Numerical id

8

Author

cvalente (11260)

Entry type

Theorem

Classification

msc 16-00

Classification

msc 20-00

Classification

msc 13-00

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

LaTeXML
Cartesian product of vector spaces

Cartesian product of vector spaces

Suppose

V

1

,

…

,

V

N

are

vector spaces

over a field

𝔽

.
Then the

Cartesian product

V

1

×

⋯

×

V

N

is a vector space
when addition and

scalar multiplication

is defined as follows

(

u

1

,

…

,

u

N

)

+

(

v

1

,

…

,

v

N

)

=

(

u

1

+

v

1

,

…

,

u

N

+

v

N

)

,

k

⁢

(

u

1

,

…

,

u

N

)

=

(

k

⁢

u

1

,

…

,

k

⁢

u

N

)

for

u

i

,

v

i

∈

V

i

,

k

∈

𝔽

.

For example, the vector space structure of

ℝ

n

if defined as above.

Properties

1.

If

V

i

are vector spaces and

W

i

⊂

V

i

are

subspaces

,
then

W

1

×

⋯

×

W

N

is a vector subspace of

V

1

×

⋯

×

V

N

.

2.

The

dimension

of

V

1

×

⋯

×

V

N

is

dim

⁡

V

1

+

⋯

+

dim

⁡

V

N

.

Title

Cartesian product of vector spaces

Canonical name

CartesianProductOfVectorSpaces

Date of creation

2013-03-22 15:16:06

Last modified on

2013-03-22 15:16:06

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

8

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 16-00

Classification

msc 13-00

Classification

msc 20-00

Classification

msc 15-00

Generated on Fri Feb 9 18:34:46 2018 by

LaTeXML
concepts in linear algebra

concepts in linear algebra

The aim of this entry is to present a list of the key
objects and

operators

used in

linear algebra

. Each entry in the
list links (or will link in the future) to the corresponding PlanetMath
entry where the object is presented in greater detail. For convenience,
this list also presents the encouraged notation to use (at PlanetMath)
for these objects.

Some of this notation is simply an example of more general notation, either notation in

set theory

or notation for

functions

. Some notation is also standard from

category theory

.

Suppose

V

is a

vector space

over a field

K

. Where the field

K

is clear from

context

it is sometimes eliminated from the notation. Let

L

be a

linear operator

, or linear transformation, from

V

to

W

, and

E

be an

endomorphism

of

V

.

•

ℒ

⁢

(

V

,

W

)

, the set of linear transformations between vector spaces

V

and

W

(also

Hom

K

⁡

(

V

,

W

)

),

•

basis of a vector space and the matrix associated with the basis,

•

dim

K

⁡

V

,

dimension

(

http://planetmath.org/Dimension2

) of

V

,

•

span

⁡

{

e

1

,

…

,

e

n

}

vector space spanned by vectors

{

e

i

}

(note that the list of vectors need not be finite). Some other notations are

(

e

1

,

e

2

,

…

,

e

n

)

or

⟨

e

1

,

e

2

,

…

,

e

n

⟩

(do not confuse last one with

similar

inner product

notation),

•

det

⁡

E

,

determinant

of a linear operator,

•

tr

⁡

E

, trace of a linear operator (also

trace

⁡

E

),

•

im

⁡

L

, image of a linear operator (also

img

⁡

L

and

L

⁢

(

V

)

),

•

ker

⁡

L

, kernel of a linear operator,

•

for sets

A

,

B

and a point

x

∈

V

, the expressions

A

+

B

,

A

-

B

,

A

+

x

are the

Minkowski sums

(

http://planetmath.org/MinkowskiSum2

) (not especially if

A

and

B

are

subspaces

, then their sum is a subspace, but the result may not be a

direct sum

),

•

V

∗

,

dual space

of a vector space

V

(also

V

∨

or

Hom

K

⁡

(

V

,

K

)

),

•

A

∗

, adjoint operator of a linear operator,

•

V

⊕

W

, direct sum of vector spaces

V

and

W

(both internal end external),

•

V

⊗

K

W

,

tensor product

of

V

and

W

, and

•

V

∧

W

, antisymmetrized tensor product (also called the

wedge product

),

•

bilinear forms

and

quadratic forms

•

generalizations

of vector spaces over a field to vector spaces over a

division ring

to modules over a ring.

Title

concepts in linear algebra

Canonical name

ConceptsInLinearAlgebra

Date of creation

2013-03-22 14:13:30

Last modified on

2013-03-22 14:13:30

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

13

Author

matte (1858)

Entry type

Definition

Classification

msc 16-00

Classification

msc 13-00

Classification

msc 20-00

Classification

msc 15-00

Generated on Fri Feb 9 18:34:48 2018 by

LaTeXML
direct sum

direct sum

Let

{

X

i

:

i

∈

I

}

be a

collection

of modules
in some

category

of modules.
Then the

direct sum

∐

i

∈

I

X

i

of that collection is the

submodule

of the

direct product

(

http://planetmath.org/DirectProduct

) of the

X

i

consisting of all

elements

(

x

i

)

such that all but a finite number
of the

x

i

are zero.

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 sum

∐

i

∈

I

X

i

satisfies

a certain

universal property

.
Namely, if

Y

is a module
and there exist

homomorphisms

f

i

:

Y

→

X

i

for all

i

∈

I

,
then there exists a unique homomorphism

ϕ

:

∐

i

∈

I

X

i

→

Y

satisfying

p

i

⁢

ϕ

=

f

i

for all

i

∈

I

.

\xymatrix

X

i

&

&

Y

\ar

[

l

l

]

f

i

&

∐

i

∈

I

X

i

\ar

[

u

l

]

p

i

\ar

@

-

-

>

[

u

r

]

ϕ

The direct sum is often referred to
as the

weak direct sum

or simply the

sum

.

Compare this to the direct product of modules.

Often an internal direct sum is written as

⊕

i

∈

I

X

i

.

Title

direct sum

Canonical

name

DirectSum

Date of creation

2013-03-22 12:09:37

Last modified on

2013-03-22 12:09:37

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

10

Author

antizeus (11)

Entry type

Definition

Classification

msc 16-00

Synonym

weak direct sum

Related topic

CategoricalDirectSum

Related topic

DirectSummand

Related topic

DirectSumOfMatrices

Generated on Fri Feb 9 18:34:50 2018 by

LaTeXML
dual module

dual module

Let

R

be a ring and

M

be a left

http://planetmath.org/node/365

R

-module. The

dual module

of

M

is
the right

http://planetmath.org/node/365

R

-module consisting of all

module homomorphisms

from

M

into

R

.

It is denoted by

M

∗

. The elements of

M

∗

are called

linear functionals

.

The action of

R

on

M

∗

is given by

(

f

⁢

r

)

⁢

(

m

)

=

(

f

⁢

(

m

)

)

⁢

r

for

f

∈

M

∗

,

m

∈

M

, and

r

∈

R

.

If

R

is

commutative

, then every

M

is an

http://planetmath.org/node/987

(

R

,

R

)

-

bimodule

with

r

⁢

m

=

m

⁢

r

for all

r

∈

R

and

m

∈

M

. Hence, it makes sense to ask whether

M

and

M

∗

are

isomorphic

. Suppose that

b

:

M

×

M

→

R

is a

bilinear form

. Then it is easy to check that for a fixed

m

∈

M

, the function

b

⁢

(

m

,

-

)

:

M

→

R

is a module homomorphism,
so is an element of

M

∗

. Then we have a module homomorphism from

M

to

M

∗

given by

m

↦

b

⁢

(

m

,

-

)

.

Title

dual module

Canonical

name

DualModule

Date of creation

2013-03-22 16:00:26

Last modified on

2013-03-22 16:00:26

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

10

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 16-00

Related topic

Unimodular

Defines

linear functional

Generated on Fri Feb 9 18:34:52 2018 by

LaTeXML
exact sequence

exact sequence

If we have two

homomorphisms

f

:

A

→

B

and

g

:

B

→

C

in some

category

of modules,
then we say that

f

and

g

are

exact

at

B

if the image of

f

is equal to the kernel of

g

.

A

sequence

of homomorphisms

⋯

→

A

n

+

1

⟶

f

n

+

1

A

n

⟶

f

n

A

n

-

1

→

⋯

is said to be

exact

if each pair
of

adjacent

homomorphisms

(

f

n

+

1

,

f

n

)

is exact –
in other words if

im

⁢

f

n

+

1

=

ker

⁢

f

n

for all

n

.

Compare this to the notion of a

chain complex

.

Remark

. The notion of

exact sequences

can be generalized to any

abelian category

𝒜

, where

A

i

and

f

i

above are objects and

morphisms

in

𝒜

.

Title

exact sequence

Canonical name

ExactSequence

Date of creation

2013-03-22 12:09:27

Last modified on

2013-03-22 12:09:27

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

7

Author

antizeus (11)

Entry type

Definition

Classification

msc 16-00

Related topic

ExactSequence2

Related topic

CategoricalSequence

Related topic

HomologicalComplexOfTopologicalVectorSpaces

Related topic

CategoricalDiagramsAsFunctors

Related topic

SpinGroup

Related topic

AlternativeDefinitionOfAnAbelianCategory

Generated on Fri Feb 9 18:34:54 2018 by

LaTeXML
examples of modules

examples of modules

This entry is a of

examples of modules

over rings. Unless otherwise specified in the example,

M

will be a module over a ring

R

.

•

Any

abelian group

is a module over the

ring of integers

, with action defined by

n

⋅

g

for

g

∈

G

given by

n

⋅

g

=

∑

i

=

1

n

g

.

•

If

R

is a subring of a ring

S

, then

S

is an

R

-module, with action given by

multiplication

in

S

.

•

If

R

is any ring, then any (left) ideal

I

of

R

is a (left)

R

-module, with action given by the multiplication in

R

.

•

Let

R

=

ℤ

and let

E

=

{

2

⁢

k

∣

k

∈

ℤ

}

. Then

E

is a module over the ring

ℤ

of

integers

. Further, define the sets

B

=

E

×

E

and

C

=

E

×

{

0

}

and

D

=

{

0

}

×

E

. Then

B

,

C

, and

D

are modules over

ℤ

×

ℤ

, with action given by

a

⋅

x

=

(

a

⋅

x

1

,

a

⋅

x

2

)

if

x

=

(

x

1

,

x

2

)

even if the

product

is redefined as

a

⋅

x

1

=

0

and

a

⋅

x

2

=

0

, but now the

identity element

is

(

1

,

1

)

. However by our new product definition

a

⋅

x

=

(

a

⋅

x

1

,

a

⋅

x

2

)

=

(

0

,

0

)

even if

a

=

(

1

,

1

)

, the ring identity element originally In the more general definition of module which does not require an identity element

𝟏

in the ring and does not require

𝟏

⋅

m

=

m

for all

m

∈

M

, we observe that

𝟏

⋅

m

≠

m

in this example just constructed. (one of the purposes of this comment is to show that all modules need not be unital ones).

•

Yetter-Drinfel’d module. (

http://planetmath.org/QuantumDouble

)

Title

examples of modules

Canonical name

ExamplesOfModules

Date of creation

2013-03-22 14:36:28

Last modified on

2013-03-22 14:36:28

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

16

Author

mathcam (2727)

Entry type

Example

Classification

msc 16-00

Classification

msc 20-00

Classification

msc 13-00

Related topic

QuantumDouble

Related topic

Module

Generated on Fri Feb 9 18:34:56 2018 by

LaTeXML
inverses in rings

inverses in rings

Let

R

be a

ring with unity

1

and

r

∈

R

. Then

r

is

left invertible

if there exists

q

∈

R

with

q

⁢

r

=

1

;

q

is a

left inverse

of

r

. Similarly,

r

is

right invertible

if there exists

s

∈

R

with

r

⁢

s

=

1

;

s

is a

right inverse

of

r

.

Note that, if

r

is left invertible, it may not have a unique left inverse, and similarly for right invertible elements. On the other hand, if

r

is left invertible and right invertible, then it has exactly one left inverse and one right inverse. Moreover, these two are equal, and

r

is a unit.

Title

inverses in rings

Canonical name

InversesInRings

Date of creation

2013-03-22 17:08:55

Last modified on

2013-03-22 17:08:55

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

4

Author

Wkbj79 (1863)

Entry type

Topic

Classification

msc 16-00

Related topic

Klein4Ring

Related topic

LeftAndRightUnityOfRing

Defines

left invertible

Defines

right invertible

Defines

left inverse

Defines

right inverse

Generated on Fri Feb 9 18:35:02 2018 by

LaTeXML