Let A be a finite-dimensional associative algebra with identity over a field k, M an A-module which is finite-dimentional as a vector space over k, and E = Horn*; (M, M) the algebra of linear transformations on M. For aeA. Let a L denote the linear transformation of M given by a Σ (x) = ax, for x e M. Define the following subalgebras of E:A L = {a L : aeA} C = {feE: f{ax) = af(x) for each ae A, x e M} D = {fe E: f{g(x)) = g{f(x)) for each geC,xeM} .Clearly, A L g D. Require M to be faithful. Then A is isomorphic to, and will be identified with, A L . If A = D, it is said that the pair (A, M) has the double centralizer property.A is called a QF-1 algebra if (A, M) has the double centralizer property for each faithful ^.-module M.The following results in the theory of QFΊ algebras are obtained:1. Let A be a commutative algebra over an arbitrary field. Then A is QF-1 if and only if A is Frobenius.2. Let A be an algebra such that the simple left A-modules are one-dimensional. Suppose there exist distinct simple two-sided ideals Ai and A 2 contained in the radical of A, and primitive idempotents e and /, such that eA k f Φ 0, for k -1, 2. Then A is not QF-1.3. Let A be an algebra with the properties that the simple left A-modules are one-dimensional, and the two-sided ideal lattice of A is distributive. Then if A satisfies any one of the following conditions, it is not QF-1.(a) There exist, for r Ξ> 2, 2r distinct simple two-sided ideals A uv contained in the radical, and primitive idempotents βi u and βj v for 1 ^ u, v ^ r, satisfying ei u A uv Ej v Φ 0, where the index pair (u, v) ranges over the set (1,1), (2,1), (2, 2), (3, 2), (3, 3), • , (r, r -1), (r, r), (1, r) .(b) There exist, for r ^ 1, 2r -f 2 distinct simple two-sided ideals A uv and Aζ, for (u, v) = (1,1), (1, 2), , (r -1, r -1), (r -1, r), and (/>, v) = (1,1), (2,1), (3, r), and (4, r), and primitive idempotents e, tt , e jv , and e /c^ satisfying e iu A uv ej Φ 0 and ejfcpAίe^ =£ 0, where (u, v) and (p, v) range over the index pairs indicated above.It is to be noted that the condition given in 2b is but one of three conditions of that type which may be formulated. An algebra 81 82 DENIS RAGAN FLOYD satisfying either of the other two conditions is also not QF-1.A special case of (2b) is worth mentioning, namely the case where the set of index pairs (u, v) which occur in statement is empty. There are two variants of the case, rather than the usual three. This special case appears separately in the following form: let A be an algebra whose simple two-sided ideals are one-dimensional, and whose two-sided ideal lattice is distributive. Suppose that either (i) e k A k e Φ 0 or (ii) eA k e k Φ 0, for k -1, 2, 3, 4, where the A k are distinct simple two-sided ideals, and the e k and e are primitive idempotents of A. Then A is not QF-l.The results (2a) and (2b) appear in Chapter 3, and are stated there in terms which involve the notion of the graph associated with the zero ideal of an algebra. The notion of the graph associated with A θ1 where A o is a two-si...
Abstract. If / is the ideal generated by all associators, (a, b, c) = (ab)ca{bc), it is well known that in any nonassociative algebra R, I C (R, R, R) + R(R, R, R). We examine nonassociative algebras where / C (R, R, R). Such algebras include (-1, 1) algebras, Lie algebras, and, as we show, a large number of associator dependent algebras. An alternator is an associator of the type (a, a, b), {a, b, a), (f>, a, a). We next study algebras where the additive span of all alternators is an ideal. These include all algebras where I = {R, R, R) as well as alternative algebras. The last section deals with prime, right alternative, alternator ideal algebras satisfying an identity of the form [x, (x, x, a)] = y(x, x, [x, a]) for fixed y. With two exceptions, if this algebra has an idempotent e such that (e, e, R) = 0, then the algebra is alternative. All our work deals with algebras with an identity element over a field of characteristic prime to 6. All our containment relations are given by identities.
R). Such algebras include (-1, 1) algebras, Lie algebras, and, as we show, a large number of associator dependent algebras. An alternator is an associator of the type (a, a, b), {a, b, a), (f>, a, a). We next study algebras where the additive span of all alternators is an ideal. These include all algebras where I = {R, R, R) as well as alternative algebras. The last section deals with prime, right alternative, alternator ideal algebras satisfying an identity of the form [x, (x, x, a)] = y(x, x, [x, a]) for fixed y. With two exceptions, if this algebra has an idempotent e such that (e, e, R) = 0, then the algebra is alternative. All our work deals with algebras with an identity element over a field of characteristic prime to 6. All our containment relations are given by identities.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.