polynomial identity algebra

polynomial identity algebra

Let

R

be a

commutative ring

with 1. Let

X

be a

countable set

of

variables

, and let

R

⁢

⟨

X

⟩

denote the

free associative algebra

over

R

. If

X

is finite, we can also write

R

⁢

⟨

X

⟩

as

R

⁢

⟨

x

1

,

…

⁢

x

n

⟩

, where the

x

i

′

⁢

s

∈

X

. Because of the freeness condition on the

algebra

, the variables are non-commuting among themselves. However, the variables do commute with elements of

R

. A typical

element

f

of

R

⁢

⟨

X

⟩

is a

polynomial

over

R

in

n

(finite) non-commuting variables of

X

.

Definition

. Let

A

be a

R

-algebra and

f

=

f

⁢

(

x

1

,

…

,

x

n

)

∈

R

⁢

⟨

X

⟩

. For any

a

1

,

…

,

a

n

∈

A

,

f

⁢

(

a

1

,

…

,

a

n

)

∈

A

is called an

evaluation of

f

at

n

-tuple

(

a

1

,

…

,

a

n

)

. If the evaluation vanishes (=0) for all

n

-tuples of

Π

i

=

1

n

⁢

A

, then

f

is called a

polynomial identity

for

A

.

A polynomial

f

∈

R

⁢

⟨

X

⟩

is

proper

, or

monic

, if, in the

homogeneous component

of the highest degree in

f

, one of its

monomials

has

coefficient

= 1.

Definition

. An algebra

A

over a commutative ring

R

is said to be a

polynomial identity algebra

over

R

, or a PI-

algebra

over

R

, if there is a proper polynomial

f

∈

R

⁢

⟨

x

1

,

…

,

x

n

⟩

, such that

f

is a polynomial identity for

A

. A

polynomial identity ring

, or PI-

ring

,

R

is a polynomial identity

ℤ

-algebra.

Examples

1.

A commutative ring is a PI-ring, satisfying the polynomial

[

x

,

y

]

=

x

⁢

y

-

y

⁢

x

.

2.

A finite field (with

q

elements) is a PI-ring, satisfying

x

q

-

x

.

3.

The ring

T

of

upper triangular

n

×

n

matrices over a field is a PI-ring. This is true because for any

a

,

b

∈

T

,

a

⁢

b

-

b

⁢

a

is strictly upper triangular (zeros along the

diagonal

). Any

product

of

n

strictly upper triangular matrices

in

T

is 0. Therefore,

T

satisfies

[

x

1

,

y

1

]

⁢

[

x

2

,

y

2

]

⁢

⋯

⁢

[

x

n

,

y

n

]

.

4.

The ring

S

of

2

×

2

matrices over a field is a PI-ring. One can show that

S

satisfies

[

[

x

1

,

x

2

]

2

,

x

3

]

. This

identity

is called the

Hall identity

.

5.

A

subring

of a PI-ring is a PI-ring. A

homomorphic image

of a PI-ring is a PI-ring.

6.

One can show that a ring

R

with polynomial identity

x

n

-

x

is

commutative

. Thus, one sees that

x

n

-

x

and

x

⁢

y

-

y

⁢

x

, although very different (one is

homogeneous of degree

2 in 2 variables, the other one is not even

homogeneous

, in one variable of degree n), are both polynomial identities for

R

.

Title

polynomial identity algebra

Canonical name

PolynomialIdentityAlgebra

Date of creation

2013-03-22 14:20:38

Last modified on

2013-03-22 14:20:38

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16U80

Classification

msc 16R10

Synonym

PI-algebra

Synonym

algebra with polynomial identity

Defines

Hall identity

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