algebras

algebras

Let

K

be a

commutative

unital ring

(often a field) and

A

a

K

-module.
Given a

bilinear mapping

b

:

A

×

A

→

A

, we say

(

K

,

A

,

b

)

is a

K

-

algebra

. We usually write only

A

for the tuple

(

K

,

A

,

b

)

.

Remark 1

.

Many authors and applications insist on

K

as a field, or at least a

local
ring

, or a

semisimple ring

. This enables

A

to have some notion of

dimension

or rank.

This definition is a compact method to encode the property that our multiplication is

distributive

: the multiplication is additive in both

variables

translates to

(

a

+

b

)

⁢

c

=

a

⁢

c

+

b

⁢

c

,

a

⁢

(

b

+

c

)

=

a

⁢

b

+

a

⁢

c

a

,

b

,

c

∈

A

.

Furthermore, the assumption that scalars can be passed in and out of the bilinear

product

translates to

(

l

⁢

a

)

⁢

b

=

l

⁢

(

a

⁢

b

)

=

a

⁢

(

l

⁢

b

)

,

a

,

b

∈

A

,

l

∈

K

.

Perhaps the most important outcome of these two axioms of an algebra is the
opportunity to express

polynomial

like

equations

over the algebra. Without the distributive axiom we cannot establish

connections

between addition and multiplication. Without scalar multiplication we cannot describe

coefficients

.
With these equations we can define certain

subalgebras

, for example we
see both axioms at work in

Proposition 2

.

Given an algebra

A

, the set

Z

0

⁢

(

A

)

=

{

z

∈

A

:

z

⁢

a

=

a

⁢

z

,

a

∈

A

}

.

Z

0

⁢

(

A

)

is a

submodule

of

A

.

Proof.

For now let elements of

A

be denoted with

a

^

to distinguish them
from scalars. As a module

0

⁢

a

^

=

0

^

for all

a

∈

A

. Then

0

^

⁢

a

^

=

(

0

⁢

a

^

)

⁢

a

^

=

(

a

^

)

⁢

(

0

⁢

a

^

)

=

a

^

⁢

0

^

.

So

0

^

∈

Z

0

⁢

(

A

)

.

Also given

z

^

,

w

^

∈

Z

0

⁢

(

A

)

then for all

a

∈

A

,

(

z

^

+

w

^

)

⁢

a

^

=

z

^

⁢

a

^

+

w

^

⁢

a

^

=

a

^

⁢

z

^

+

a

^

⁢

w

^

=

a

^

⁢

(

z

^

+

w

^

)

.

So

z

^

+

w

^

∈

A

.

Finally, given

l

∈

K

we have

(

l

⁢

z

^

)

⁢

a

^

=

l

⁢

(

z

^

⁢

a

^

)

=

l

⁢

(

a

^

⁢

z

^

)

=

a

^

⁢

(

l

⁢

z

^

)

.

∎

Although this set

Z

⁢

(

A

)

appears like a reasonable object to define as the
center of an algebra, it is usually preferable to produce a subalgebra, not
simply a submodule, and for this we need elements that can be regrouped in products associatively, that is, that lie in the nucleus. So the center is commonly defined as

Z

⁢

(

A

)

=

{

z

∈

A

:

z

⁢

a

=

a

⁢

z

,

z

⁢

(

a

⁢

b

)

=

(

z

⁢

a

)

⁢

b

,

a

⁢

(

z

⁢

b

)

=

(

a

⁢

z

)

⁢

b

,

(

a

⁢

b

)

⁢

z

=

a

⁢

(

b

⁢

z

)

,

a

,

b

∈

A

}

.

When the algebra

A

has an

identity

(unity)

1

then we can go further to identify

K

as a subalgebra of

A

by

l

⁢

1

. Then we see this subalgebra is
necessarily in the center of

A

. As a converse, given a unital ring

R

(

associativity

is necessary), the center of the ring forms a commutative unital

subring

over which

R

is an algebra. In this way unital rings and associative unital algebras are often interchanged.

Title

algebras

Canonical name

Algebras

Date of creation

2013-03-22 16:27:20

Last modified on

2013-03-22 16:27:20

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

7

Author

Algeboy (12884)

Entry type

Definition

Classification

msc 17A01

Related topic

ring

Related topic

NonAssociativeAlgebra

Related topic

FreeAssociativeAlgebra

Related topic

TopicEntryOnTheAlgebraicFoundationsOfMathematics

Defines

algebra

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