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a nonembedding theorem for algebras - American Mathematical Society

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Proof. Suppose to the contrary that B is an ideal of A, B C$>(A(a, Я)) and r2 C Ma, Я, r, a). Let L be the mapping of A onto E(r2, A) given by. L : a —» La restricted.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 50, July 1975

A NONEMBEDDINGTHEOREM FOR ALGEBRAS ERNEST L. STITZINGER ABSTRACT.

for Lie eral

A certain

and associative

other

consequences

Recently,

are

D. A. Towers

associative

algebra.

associative

algebras.

The

special

Let

L

has

results

has

gives

considered

previously

been

a generalization

the Frattini

had been

previously

of the present

which

shown

and

sev-

appear

subalgebra

of a non-

considered

note

for Lie and

is to give

a generaliza-

in [7] and [8] and to observe

cases.

A be a nonassociative

restricted

A by letting

ciative

note

noted.

The purpose

the left multiplication

= \L

result

This

concept

tion of the nonembedding some

nonembedding

algebras.

algebra

F.

For

of A by x. For any left ideal

to C, for all

x e A ¡. For

a o b = a(ab)

+ ß(ba)

by A(a,

For

algebra

over a field

ß).

for all

x e A, denote

C of A, let E(C, A)

a, ß € F, define

a new product

a, b e A and denote

a, ß, r, a £ F,

by

this

in

nonasso-

let

/V(a, ß, r, a) = \x e A; (L aob , - rL abL, - oL.L ' b

cl

)x = 0 for all a, b € A\.

If C C Ma, ß, T, a), then in E(C, A) define

L a o L,b = rL abL, + oL.L = L aob „, b a This

is a well-defined

homomorphism as an algebra

product

from A(a, under

ß)

this

in into

product.

for all a, b £ A.

E(C,

A) and the mapping

E(C,

A), Henceforth

Now let

B be an ideal

fj = \x £ B; L (x) = 0 for all y £ B\ and define

L (x) £ r._.

for any algebra

D,

of D if maximal

If 77 is a homomorphism Received

in A and define

r. inductively

by r. = \x e B;

= r-1(Lxoy-oLxLy)(z)£r._1. let

(D) be the intersection

subalgebras

and phrases.

of D exist

from D, then

by the editors

Frattini

of all maximal

and let

sub-

*I>(D) = D otherwise.

n((D)) C ^(n(D)).

March 11, 1974 and, in revised

AMS(MOS)subject classifications Key words

A)

if z £ r., x £ A, y £ B, then Ly(xz)

algebras

is a

E(C,

for all y £ B \. If r ¿ 0 and r. C Ma, ß, r, a), then r. is a left

ideal of A since

Finally

L : a —» L

we consider

form, May 2, 1974.

(1970). Primary 17A99, 17E05. subalgebra. Copyright © 1975. American Mathematical Society

10 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

A NONEMBEDDING THEOREM FOR ALGEBRAS Theorem.

Let

B be a nonassociative

algebra

11

over a field

F such

that

dim r. = 1 and 1 < dim r. < (A(a,

of A onto

L is a homomorphism

E(r2,

ß))

A) given

of A(a,

ß)

by

onto

E(r2, A) and

E(r2, B) = L(ß) Ç L(«A(a, Let

z

••• , z

be a basis

i = 1, • • • , k - 1, let

j8))) Ç