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 â» 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
