explicit definition of polynomial rings in arbitrarly many variables

explicit definition of polynomial rings in arbitrarly many variables

Let

R

be a ring and let

𝕏

be any set (possibly empty). We wish to give an explicit and formal definition of the

polynomial ring

R

⁢

[

𝕏

]

.

We start with the set

ℱ

(

𝕏

)

=

{

f

:

𝕏

→

ℕ

|

f

(

x

)

=

0

for almost all

x

}

.

If

𝕏

=

{

X

1

,

…

,

X

n

}

then the elements of

ℱ

⁢

(

𝕏

)

can be interpreted as

monomials

X

1

α

1

⁢

⋯

⁢

X

n

α

n

.

Now define

R

[

𝕏

]

=

{

F

:

ℱ

(

𝕏

)

→

R

|

F

(

f

)

=

0

for almost all

f

}

.

The

addition

in

R

⁢

[

𝕏

]

is defined as

pointwise addition

.

Now we will define

multiplication

. First note that we have a multiplication on

ℱ

⁢

(

𝕏

)

. For any

f

,

g

:

𝕏

→

ℕ

put

(

f

⁢

g

)

⁢

(

x

)

=

f

⁢

(

x

)

+

g

⁢

(

x

)

.

This is the same as multiplying

x

a

⋅

x

b

=

x

a

+

b

.

Now for any

f

∈

ℱ

⁢

(

𝕏

)

define

M

⁢

(

h

)

=

{

(

f

,

g

)

∈

ℱ

⁢

(

𝕏

)

2

|

h

=

f

⁢

g

}

,

Now if

F

,

G

∈

R

⁢

[

𝕏

]

then we define the multiplication

F

⁢

G

:

ℱ

⁢

(

𝕏

)

→

R

by putting

(

F

⁢

G

)

⁢

(

h

)

=

∑

(

f

,

g

)

)

∈

M

(

h

)

F

⁢

(

f

)

⁢

G

⁢

(

g

)

.

Note that all of this well-defined, since both

F

and

G

vanish almost everywhere.

It can be shown that

R

⁢

[

𝕏

]

with these

operations

is a ring, even an

R

-

algebra

. This algebra is

commutative

if and only if

R

is. Furthermore we have an algebra homomorphism

E

:

R

→

R

⁢

[

𝕏

]

which is defined as follows: for any

r

∈

R

let

F

r

:

ℱ

⁢

(

𝕏

)

→

R

be the

function

such that if

f

:

𝕏

→

ℕ

is such that

f

⁢

(

x

)

=

0

for any

x

∈

𝕏

, then put

F

r

⁢

(

f

)

=

r

and for any other function

f

∈

ℱ

⁢

(

𝕏

)

put

F

r

⁢

(

f

)

=

0

. Then

E

⁢

(

r

)

=

F

r

is our function, which is a

monomorphism

. Furthermore if

R

is unital with the

identity

1

, then

E

⁢

(

1

)

is the identity in

R

⁢

[

𝕏

]

. Anyway we can always interpret

R

as a subset of

R

⁢

[

𝕏

]

if put

r

=

F

r

for

r

∈

R

.

Note, that if

𝕏

=

∅

, then

R

⁢

[

∅

]

is still defined and

E

:

R

→

R

⁢

[

𝕏

]

is an

isomorphism

of rings (it is ,,onto”). Actually these two conditions are

equivalent

.

Also note, that

𝕏

itself can be interpreted as a subset of

R

⁢

[

𝕏

]

. Indeed, for any

x

∈

𝕏

define

f

x

:

𝕏

→

ℕ

by

f

x

⁢

(

x

)

=

1

and

f

x

⁢

(

y

)

=

0

for any

y

≠

x

. Then define

F

x

:

ℱ

⁢

(

𝕏

)

→

R

by putting

F

x

⁢

(

f

x

)

=

1

and

F

x

⁢

(

f

)

=

0

for any

f

≠

f

x

. It can be easily seen that

F

x

=

F

y

if and only if

x

=

y

. Thus we will use convention

x

=

F

x

.

With these notations (i.e.

R

,

𝕏

⊆

R

⁢

[

𝕏

]

) we have that elements of

R

⁢

[

𝕏

]

are exactly

polynomials

in the set of

variables

𝕏

with

coefficients

in

R

.

Title

explicit definition of polynomial rings in arbitrarly many variables

Canonical name

ExplicitDefinitionOfPolynomialRingsInArbitrarlyManyVariables

Date of creation

2013-03-22 19:18:10

Last modified on

2013-03-22 19:18:10

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

8

Author

joking (16130)

Entry type

Definition

Classification

msc 12E05

Classification

msc 13P05

Classification

msc 11C08

Generated on Fri Feb 9 16:40:19 2018 by

LaTeXML