aarjaneiro/planetmath huggingface dataset:
minus one times an element is the additive inverse in a ring

minus one times an element is the additive inverse in a ring

Lemma 1

.

Let

R

be a ring (with unity

1

) and let

a

be an element of

R

. Then

(

-

1

)

⋅

a

=

-

a

where

-

1

is the

additive inverse

of

1

and

-

a

is the additive inverse of

a

.

Proof.

Note that for any

a

in

R

there exists a unique “

-

a

” by the uniqueness of additive inverse in a ring. We check that

(

-

1

)

⋅

a

equals the additive inverse of

a

.

a

+

(

-

1

)

⋅

a

=

1

⋅

a

+

(

-

1

)

⋅

a

,

by the definition of

⁢

1

=

(

1

+

(

-

1

)

)

⋅

a

,

by the distributive law

=

0

⋅

a

,

by the definition of

-

1

=

0

,

as a result of the properties of zero

Hence

(

-

1

)

⋅

a

is “an” additive inverse for

a

, and by uniqueness

(

-

1

)

⋅

a

=

-

a

,

the

additive inverse of

a

. Analogously, we can prove that

a

⋅

(

-

1

)

=

-

a

as well.
∎

Title

minus one times an element is the additive inverse in a ring

Canonical name

MinusOneTimesAnElementIsTheAdditiveInverseInARing

Date of creation

2013-03-22 14:14:00

Last modified on

2013-03-22 14:14:00

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

9

Author

alozano (2414)

Entry type

Theorem

Classification

msc 16-00

Classification

msc 13-00

Classification

msc 20-00

Synonym

(

-

1

)

⋅

a

=

-

a

Related topic

0cdotA0

Generated on Fri Feb 9 18:35:04 2018 by

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new vector spaces from old ones

new vector spaces from old ones

This entry list methods that give new

vector spaces

from old ones.

1.

Changing the field (

complexification

, etc.)

2.

vector subspace

3.

Quotient vector space

4.

direct product

of vectors spaces

5.

Cartesian product of vector spaces

6.

Tensor product

of vector spaces (

http://planetmath.org/TensorProductClassical

)

7.

The space of

linear maps

from one vector space to another, also denoted by

Hom

k

⁡

(

V

,

W

)

, or simply

Hom

⁡

(

V

,

W

)

, where

V

and

W

are vector spaces over the field

k

8.

The space of

endomorphisms

of a vector space. Using the notation above, this is the space

Hom

k

⁡

(

V

,

V

)

=

End

⁡

(

V

)

9.

dual vector space

(

http://planetmath.org/DualSpace

), and bi-dual vector space. Using the notation above, this is the space

Hom

⁡

(

V

,

k

)

, or simply

V

*

.

10.

The

annihilator

of a

subspace

is a subspace of the dual vector space

11.

Wedge product

of vector spaces

12.

A field

k

is a vector space over itself. Consider a set

B

and the set

V

of all functions from

B

to

k

. Then

V

has a natural vector space structure. If

B

is finite, then

V

can be viewed as a vector space having

B

as a basis.

Vector spaces involving a linear map

Suppose

L

:

V

→

W

is a linear map.

1.

The kernel of

L

is a subspace of

V

.

2.

The image of

L

is a subspace of

W

.

3.

The

cokernel

of

L

is a

quotient space

of

W

.

Topological vector spaces

Suppose

V

is

topological vector space

.

1.

If

W

is a subspace of

V

then its

closure

W

¯

is also a subspace of

V

.

2.

If

V

is a metric vector space then its

completion

V

~

is also a (metric) vector space.

3.

The

direct integral of Hilbert spaces

provides a new

Hilbert space

.

Spaces of structures and subspaces of the tensor algebra of a vector space

There are also certain spaces of interesting structures on a vector
space that at least in the case of finite

dimension

correspond to
certain subspaces of the

tensor algebra

of the vector space. These
spaces include:

1.

The space of

Euclidean

inner products

.

2.

The space of

Hermitian inner products

.

3.

the space of symplectic structures.

4.

vector bundles

5.

space of

connections

Title

new vector spaces from old ones

Canonical name

NewVectorSpacesFromOldOnes

Date of creation

2013-03-22 15:31:08

Last modified on

2013-03-22 15:31:08

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

16

Author

matte (1858)

Entry type

Topic

Classification

msc 16-00

Classification

msc 13-00

Classification

msc 20-00

Classification

msc 15-00

Generated on Fri Feb 9 18:35:07 2018 by

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Noether normalization lemma

Noether normalization lemma

Let

F

be a field and

K

a finitely-generated

commutative algebra

over

F

. There there exists a non-negative

integer

n

and elements

x

1

,

…

,

x

n

∈

A

,

algebraically independent

over

F

, such that

K

is integral over

F

⁢

[

x

1

,

…

,

x

n

]

.

Title

Noether normalization lemma

Canonical name

NoetherNormalizationLemma

Date of creation

2013-03-22 14:14:36

Last modified on

2013-03-22 14:14:36

Owner

dooder0001 (4288)

Last modified by

dooder0001 (4288)

Numerical id

7

Author

dooder0001 (4288)

Entry type

Theorem

Classification

msc 16-00

Synonym

Noetherian normalization lemma

Generated on Fri Feb 9 18:35:08 2018 by

LaTeXML
quotient ring

quotient ring

Definition.

Let

R

be a ring and let

I

be a

two-sided ideal

(

http://planetmath.org/Ideal

) of

R

.
To define the

quotient ring

R

/

I

, let us first
define an

equivalence relation

in

R

. We say that the

elements

a

,

b

∈

R

are

equivalent

, written as

a

∼

b

, if and only if

a

-

b

∈

I

.
If

a

is an element of

R

, we denote the corresponding

equivalence
class

by

[

a

]

. Thus

[

a

]

=

[

b

]

if and only if

a

-

b

∈

I

.
The

quotient ring

of

R

modulo

I

is the set

R

/

I

=

{

[

a

]

|

a

∈

R

}

, with a ring

structure

defined as follows.
If

[

a

]

,

[

b

]

are equivalence classes in

R

/

I

, then

•

[

a

]

+

[

b

]

=

[

a

+

b

]

,

•

[

a

]

⋅

[

b

]

=

[

a

⋅

b

]

.

Here

a

and

b

are some elements in

R

that represent

[

a

]

and

[

b

]

.
By construction, every element in

R

/

I

has such a representative in

R

.
Moreover, since

I

is

closed under

addition

and

multiplication

, one can
verify that the ring structure in

R

/

I

is well defined.

A common notation is

a

+

I

=

[

a

]

which is

consistent

with the notion of classes

[

a

]

=

a

⁢

H

∈

G

/

H

for a group

G

and a

normal subgroup

H

.

Properties

1.

If

R

is

commutative

, then

R

/

I

is commutative.

2.

The

mapping

R

→

R

/

I

,

a

↦

[

a

]

is a

homomorphism

, and
is called the

natural homomorphism

(

http://planetmath.org/NaturalHomomorphism

).

Examples

1.

For a ring

R

, we have

R

/

R

=

{

[

0

]

}

and

R

/

{

0

}

=

R

.

2.

Let

R

=

ℤ

, and let

I

=

2

⁢

ℤ

be the set of

even numbers

.
Then

R

/

I

contains only two classes; one for even numbers,
and one for

odd numbers

. Actually this quotient ring is a field. It is the only field with two elements (up to isomorphy) and is also denoted by

𝔽

2

.

3.

One way to construct

complex numbers

is to consider the field

ℝ

⁢

[

T

]

/

(

T

2

+

1

)

. This field can viewed as the set of all

polynomials

of degree

1

with normal addition and

(

a

+

b

⁢

T

)

⁢

(

c

+

d

⁢

T

)

=

a

⁢

c

-

b

⁢

d

+

(

a

⁢

d

+

b

⁢

c

)

⁢

T

, which is like

complex multiplication

.

Title

quotient ring

Canonical name

QuotientRing

Date of creation

2013-03-22 11:52:32

Last modified on

2013-03-22 11:52:32

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

18

Author

mathwizard (128)

Entry type

Definition

Classification

msc 16-00

Classification

msc 81R12

Classification

msc 20C30

Classification

msc 81R10

Classification

msc 81R05

Classification

msc 20C32

Synonym

difference ring

Synonym

factor ring

Synonym

residue-class ring

Related topic

NaturalHomomorphism

Related topic

QuotientRingModuloPrimeIdeal

Generated on Fri Feb 9 18:35:11 2018 by

LaTeXML
ring

ring

A

ring

is a set

R

together with two binary operations, denoted

+

:

R

×

R

⟶

R

and

⋅

:

R

×

R

⟶

R

, such that

1.

(

a

+

b

)

+

c

=

a

+

(

b

+

c

)

and

(

a

⋅

b

)

⋅

c

=

a

⋅

(

b

⋅

c

)

for all

a

,

b

,

c

∈

R

(

associative

law)

2.

a

+

b

=

b

+

a

for all

a

,

b

∈

R

(

commutative law

)

3.

There exists an element

0

∈

R

such that

a

+

0

=

a

for all

a

∈

R

(

additive identity

)

4.

For all

a

∈

R

, there exists

b

∈

R

such that

a

+

b

=

0

(

additive inverse

)

5.

a

⋅

(

b

+

c

)

=

(

a

⋅

b

)

+

(

a

⋅

c

)

and

(

a

+

b

)

⋅

c

=

(

a

⋅

c

)

+

(

b

⋅

c

)

for all

a

,

b

,

c

∈

R

(

distributive law

)

Equivalently, a ring is an

abelian group

(

R

,

+

)

together with a second binary operation

⋅

such that

⋅

is associative and distributes over

+

. Additive inverses are unique, and one can define

subtraction

in any ring using the formula

a

-

b

:=

a

+

(

-

b

)

where

-

b

is the additive inverse of

b

.

We say

R

has a

multiplicative identity

if there exists an element

1

∈

R

such that

a

⋅

1

=

1

⋅

a

=

a

for all

a

∈

R

. Alternatively, one may say that

R

is a

ring with unity

, a

unital ring

, or a

unitary ring

. Oftentimes an author will adopt the convention that all rings have a multiplicative identity. If

R

does have a multiplicative identity, then a

multiplicative inverse

of an element

a

∈

R

is an element

b

∈

R

such that

a

⋅

b

=

b

⋅

a

=

1

. An element of

R

that has a multiplicative inverse is called a

unit

of

R

.

A ring

R

is

commutative

if

a

⋅

b

=

b

⋅

a

for all

a

,

b

∈

R

.

Title

ring

Canonical

name

Ring

Date of creation

2013-03-22 11:48:40

Last modified on

2013-03-22 11:48:40

Owner

djao (24)

Last modified by

djao (24)

Numerical id

19

Author

djao (24)

Entry type

Definition

Classification

msc 16-00

Classification

msc 20-00

Classification

msc 13-00

Classification

msc 81P10

Classification

msc 81P05

Classification

msc 81P99

Related topic

ExampleOfRings

Related topic

Subring

Related topic

Semiring

Related topic

Group

Related topic

Associates

Defines

multiplicative identity

Defines

multiplicative inverse

Defines

ring with unity

Defines

unit

Defines

ring addition

Defines

ring multiplication

Defines

ring sum

Defines

ring product

Defines

unital ring

Defines

unitary ring

Generated on Fri Feb 9 18:35:13 2018 by

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unitization

unitization

The operation of

unitization

allows one to add a unity element to an

algebra

. Because of this construction, one can regard any algebra as a

subalgebra

of an algebra with unity. If the algebra already has a unity, the operation creates a larger algebra in which the old unity is no longer the unity.

Let

𝐀

be an algebra over a ring

𝐑

with unity

1

. Then, as a module, the unitization of

𝐀

is the

direct sum

of

𝐑

and

𝐀

:

𝐀

+

=

𝐑

⊕

𝐀

The product operation is defined as follows:

(

x

,

a

)

⋅

(

y

,

b

)

=

(

x

⁢

y

,

a

⁢

b

+

x

⁢

b

+

y

⁢

a

)

The unity of

𝐀

+

is

(

1

,

0

)

.

It is also possible to unitize any ring using this construction if one regards the ring as an algebra over the

ring of integers

(

http://planetmath.org/

Integer

). (See the entry every ring is an integer algebra for details.) It is worth noting,
however, that the result of unitizing a ring this way will always be a ring whose unity has zero

characteristic

. If one
has a ring of finite characteristic

k

, one can instead regard it as an algebra over

ℤ

k

and unitize
accordingly to obtain a ring of characteristic

k

.

The construction described above is often called “minimal unitization”. It is in fact minimal, in the sense that
every other unitization contains this unitization as a subalgebra.

Title

unitization

Canonical name

Unitization

Date of creation

2013-03-22 14:47:36

Last modified on

2013-03-22 14:47:36

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

9

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 16-00

Classification

msc 13-00

Classification

msc 20-00

Synonym

minimal unitization

Generated on Fri Feb 9 18:35:15 2018 by

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vector space

vector space

Let

F

be a field (or, more generally, a division ring). A

vector space

V

over

F

is a set with two operations,

+

:

V

×

V

⟶

V

and

⋅

:

F

×

V

⟶

V

, such that

1.

(

𝐮

+

𝐯

)

+

𝐰

=

𝐮

+

(

𝐯

+

𝐰

)

for all

𝐮

,

𝐯

,

𝐰

∈

V

2.

𝐮

+

𝐯

=

𝐯

+

𝐮

for all

𝐮

,

𝐯

∈

V

3.

There exists an element

𝟎

∈

V

such that

𝐮

+

𝟎

=

𝐮

for all

𝐮

∈

V

4.

For any

𝐮

∈

V

, there exists an element

𝐯

∈

V

such that

𝐮

+

𝐯

=

𝟎

5.

a

⋅

(

b

⋅

𝐮

)

=

(

a

⋅

b

)

⋅

𝐮

for all

a

,

b

∈

F

and

𝐮

∈

V

6.

1

⋅

𝐮

=

𝐮

for all

𝐮

∈

V

7.

a

⋅

(

𝐮

+

𝐯

)

=

(

a

⋅

𝐮

)

+

(

a

⋅

𝐯

)

for all

a

∈

F

and

𝐮

,

𝐯

∈

V

8.

(

a

+

b

)

⋅

𝐮

=

(

a

⋅

𝐮

)

+

(

b

⋅

𝐮

)

for all

a

,

b

∈

F

and

𝐮

∈

V

Equivalently, a vector space is a module

V

over a ring

F

which is a field (or, more generally, a division ring).

The elements of

V

are called

vectors

, and the element

𝟎

∈

V

is called the

zero vector

of

V

.

Title

vector space

Canonical name

VectorSpace

Date of creation

2013-03-22 11:49:10

Last modified on

2013-03-22 11:49:10

Owner

djao (24)

Last modified by

djao (24)

Numerical id

17

Author

djao (24)

Entry type

Definition

Classification

msc 16-00

Classification

msc 13-00

Classification

msc 20-00

Classification

msc 15-00

Classification

msc 70B15

Synonym

linear space

Related topic

Module

Related topic

Vector2

Related topic

Vector

Related topic

VectorSubspace

Defines

zero vector

Generated on Fri Feb 9 18:35:17 2018 by

LaTeXML
zero vector in a vector space is unique

zero vector in a vector space is unique

Theorem

The

zero vector

in a

vector space

is unique.

Proof.

Suppose

0

and

0

~

are zero vectors
in a vector space

V

. Then both

0

and

0

~

must satisfy
axiom 3 (

http://planetmath.org/VectorSpace

),
i.e., for all

v

∈

V

,

v

+

0

=

v

,

v

+

0

~

=

v

.

Setting

v

=

0

~

in the first

equation

, and

v

=

0

in the second yields

0

~

+

0

=

0

~

and

0

+

0

~

=

0

. Thus, using
axiom 2 (

http://planetmath.org/VectorSpace

),

0

=

0

~

+

0

=

0

+

0

~

=

0

~

,

and

0

=

0

~

.

□

Title

zero vector in a vector space is unique

Canonical name

ZeroVectorInAVectorSpaceIsUnique

Date of creation

2013-03-22 13:37:16

Last modified on

2013-03-22 13:37:16

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

7

Author

matte (1858)

Entry type

Theorem

Classification

msc 16-00

Classification

msc 13-00

Classification

msc 20-00

Classification

msc 15-00

Related topic

IdentityElementIsUnique

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

LaTeXML
order (of a ring)

order (of a ring)

The

order

of a ring

R

is the order (

http://

planetmath

.org/OrderGroup

) of its

additive group

, i.e. (

http://planetmath.org/Ie

) the number of elements of

R

. The order of

R

can be denoted as

|

R

|

. If

|

R

|

is finite, then

R

is said to be a

finite ring

.

This definition of order is not necessarily standard. Please see

http://planetmath.org/?op=getobj&from=corrections&id=12149

this correction and the posts attached to it for more details.

This definition of order is used in the following works:

1.

Angerer, Josef and Pilz, Günter. “The Structure of Near Rings of Small Order.”

Computer Algebra

: EUROCAM ’82, European Computer Algebra Conference; Marseilles, France, April 1982

. Editors: Goos, G. and Hartmanis, J. Berlin: Springer-Verlag, 1982, pp. 57-64.

2.

Buck, Warren.

http://planetmath.org/?op=getobj&from=papers&id=336

Cyclic Rings

. Charleston, IL: Eastern Illinois University, 2004.

3.

Fine, Benjamin. “Classification of Finite Rings of Order

p

2

.”

Mathematics Magazine

, vol. 66 #4. Washington, DC:

Mathematical Association of America

, 1993, pp. 248-252.

4.

Fletcher, Colin R. “Rings of Small Order.”

The Mathematical Gazette

, vol. 64 #427. Leicester, England: The Mathematical Association, 1980, pp. 9-22.

5.

Lam, Tsi-Yuen.

A First Course in Noncommutative Rings

. New York: Springer-Verlag, 2001.

6.

Mitchell, James.

School of Mathematics and Statistics: MT4517 Rings and Fields, Lecture Notes 1

. St. Andrews, Scotland: University of St. Andrews, 2006. URL:

http://www-history.mcs.st-and.ac.uk/ jamesm/teaching/MT4517/MT4517-notes1.pdf

http://www-history.mcs.st-and.ac.uk/ jamesm/teaching/MT4517/MT4517-notes1.pdf

7.

Nöbauer, Christof.

Numbers of rings on groups of prime power order

. Linz, Austria: Johannes Kepler Universität Linz. URL:

http://www.

algebra

.uni-linz.ac.at/ noebsi/ringtable.html

http://www.algebra.uni-linz.ac.at/ noebsi/ringtable.html

8.

Schwabe, Eric J. and Sutherland, Ian M. “Efficient Mappings for Parity-Declustered Data Layouts.”

Computing and Combinatorics: 9th Annual International Conference, COCOON 2003; Big Sky, MT, USA, July 2003; Proceedings

. Editors: Warnow, Tandy and Zhu, Binhai. Berlin: Springer-Verlag, 2003, pp. 252-261.

Title

order (of a ring)

Canonical name

OrderofARing

Date of creation

2013-03-22 17:10:31

Last modified on

2013-03-22 17:10:31

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

15

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 16-01

Synonym

order

Synonym

order of a ring

Related topic

OrderGroup

Defines

finite ring

Generated on Fri Feb 9 18:35:21 2018 by

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additive inverse of a sum in a ring

additive inverse of a sum in a ring

Let

R

be a ring with elements

a

,

b

∈

R

.
Suppose we want to find the

inverse

of the element

(

a

+

b

)

∈

R

.
(Note that we call the element

(

a

+

b

)

the sum of

a

and

b

.)
So we want the unique element

c

∈

R

so that

(

a

+

b

)

+

c

=

0

.
Actually, let’s put

c

=

(

-

a

)

+

(

-

b

)

where

(

-

a

)

∈

R

is the

additive inverse

of

a

and

(

-

b

)

∈

R

is the additive inverse of

b

.
Because addition in the ring is both

associative

and

commutative

we see that

(

a

+

b

)

+

(

(

-

a

)

+

(

-

b

)

)

=

(

a

+

(

-

a

)

)

+

(

b

+

(

-

b

)

)

=

0

+

0

=

0

since

(

-

a

)

∈

R

is the additive inverse of

a

and

(

-

b

)

∈

R

is the additive inverse of

b

.
Since additive inverses are unique this means that the additive inverse of

(

a

+

b

)

must be

(

-

a

)

+

(

-

b

)

. We write this as

-

(

a

+

b

)

=

(

-

a

)

+

(

-

b

)

.

It is important to note that we

cannot

just distribute the minus sign across the sum because this would imply that

-

1

∈

R

which is not the case if our ring is not with unity.

Title

additive inverse of a sum in a ring

Canonical name

AdditiveInverseOfASumInARing

Date of creation

2013-03-22 15:45:02

Last modified on

2013-03-22 15:45:02

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

11

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 16B70

Generated on Fri Feb 9 18:35:26 2018 by

LaTeXML
additive inverse of an inverse element

additive inverse of an inverse element

In any ring

R

, the

additive inverse

of an element

a

∈

R

must exist, is unique and is denoted by

-

a

. Since

-

a

is also in the ring

R

it

also

has an additive inverse in

R

, which is

-

(

-

a

)

. Put

-

(

-

a

)

=

c

∈

R

.
Then by definition of the additive inverse,

-

a

+

c

=

0

and

-

a

+

a

=

0

. Since additive inverses are unique, it must be that

c

=

a

.

Title

additive inverse of an inverse element

Canonical name

AdditiveInverseOfAnInverseElement

Date of creation

2013-03-22 15:45:16

Last modified on

2013-03-22 15:45:16

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

8

Author

Mathprof (13753)

Entry type

Result

Classification

msc 16B70

Related topic

InverseOfInverseInAGroup

Generated on Fri Feb 9 18:35:28 2018 by

LaTeXML
additive inverse of the zero in a ring

additive inverse of the zero in a ring

In any ring

R

, the

additive identity

is unique and usually denoted by

0

. It is called the zero or

neutral element

of the ring and it

satisfies

the zero

property

under

multiplication

. The

additive inverse

of the zero must be zero itself. For suppose otherwise: that there is some nonzero

c

∈

R

so that

0

+

c

=

0

. For any

element

a

∈

R

we have

a

+

0

=

a

since

0

is the additive identity. Now, because

addition

is

associative

we have

0

=

a

+

0

=

a

+

(

0

+

c

)

=

(

a

+

0

)

+

c

=

a

+

c

.

Since

a

is any arbitrary element in the ring, this would imply that (nonzero)

c

is an additive identity, contradicting the uniqueness of the additive identity. And so our suppostition that

0

has a nonzero

inverse

cannot be true. So the additive inverse of the zero is zero itself. We can write this as

-

0

=

0

, where the

-

sign means “additive inverse”.

Yes, for sure, there are other ways to come to this result, and we encourage you to have a bit of fun describing your own reasons for why the additive inverse of the zero of the ring must be zero itself.

For example, since

0

is the neutral element of the ring this means that

0

+

0

=

0

. From this it immediately follows that

-

0

=

0

.

Title

additive inverse of the zero in a ring

Canonical

name

AdditiveInverseOfTheZeroInARing

Date of creation

2013-03-22 15:45:13

Last modified on

2013-03-22 15:45:13

Owner

aplant (12431)

Last modified by

aplant (12431)

Numerical id

9

Author

aplant (12431)

Entry type

Definition

Classification

msc 16B70

Generated on Fri Feb 9 18:35:29 2018 by

LaTeXML
characteristic of finite ring

characteristic of finite ring

The

characteristic

(

http://planetmath.org/Characteristic

) of the

residue class ring

ℤ

/

m

⁢

ℤ

, which contains

m

elements, is

m

, too.  More generally, one has the

Theorem.

The characteristic of a

finite ring

divides the number of the elements of the ring.

Proof.

Let

n

be the characteristic of the ring

R

with

m

elements.  Since

m

is the order (

http://planetmath.org/OrderGroup

) of the group

(

R

,

+

)

,  the Lagrange’s theorem implies that

m

⁢

a

=

0

∀

a

∈

R

.

Let

m

=

q

⁢

n

+

r

where

0

≦

r

<

n

.  Because

r

⁢

a

=

(

m

-

q

⁢

n

)

⁢

a

=

m

⁢

a

-

q

⁢

(

n

⁢

a

)

=

0

-

0

=

0

∀

a

∈

R

and

n

is the least

positive integer

ν

making all

ν

⁢

a

=

0

,  the number

r

must vanish.  Therefore,

m

=

q

⁢

n

,  i.e.

n

∣

m

.

Remark.

A ring

R

, the

polynomial ring

R

⁢

[

X

]

and the ring

R

⁢

[

[

X

]

]

of

formal power series

have always the same characteristic.

Title

characteristic of finite ring

Canonical name

CharacteristicOfFiniteRing

Date of creation

2013-03-22 19:10:19

Last modified on

2013-03-22 19:10:19

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Theorem

Classification

msc 16B99

Related topic

Multiple

Related topic

IdealOfElementsWithFiniteOrder

Generated on Fri Feb 9 18:35:32 2018 by

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