Power-associative algebras in which every subalgebra is an ideal

Abstract

By an iί-algebra we mean a nonassociative algebra (not necessarily finite-dimensional) over a field in which every subalgebra is an ideal of the algebra. In this paper we prove MAIN THEOREM. Let A be a power-associative algebra over a field F of characteristic not 2. A is an iί-algebra if and only if A is one of the following; (1) a one-dimensional idempotent algebra; (2) a zero algebra; (3) an algebra with basis u Q , Ui, iel (an index set of arbitrary cardinality) satisfying UiUj = a^Uo, <*ij € F, i, j e I, all other products zero. Moreover if J is a finite subset of ί, then 'Σu.jejaijXiXj is nondegenerate in that ^a>jej a n^aj-0, a if ajGF, ieJ implies «$ = (), ie J; (4) direct sums of algebras of types (1), (2), (3) with at most one from each.

Power-Associative Algebras in which Every Subalgebra is a Left Ideal

“…Also, 6, x, x 2 are linearly independent. Now {e+x+x 2 } is one-dimensional but e+x = e(e + x + x 2 ) and x 2 We conclude that A e Q) # 0 implies A e (0)=0. If ^e(i)=0 then either ^4={e} which has basis S a for a= 1 or A={e}@ A e (Q) where >4 e (0) is a nil L-algebra.…”

“…Consider {x-ny}. We have (x-ny) 2 =x 2 -nxy-nyx + /2 2 y 2 = 0. Therefore {x -ny} is one-dimensional.…”

“…An //-algebra is a power-associative algebra in which every subalgebra is an ideal. The //-algebras were characterized by D. L. Outcalt in [2]. Let S a be the semigroup with cardinality a such that if x,yeS a then xy=y.…”

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“…In the same framework, the structure of associative, alternative and Jordan algebras in which every subalgebra is an ideal was discussed by S.-H. Liu in [11], and the same question for power-associative algebras was considered by D.L. Outcalt in [12]. Now, it is immediate to see that a Lie algebra in which all subalgebras are ideals is necessarily abelian.…”