Some Ring Theorems With Applications to Modular Representations

Nesbitt, Cecil J.; Thrall, Robert M.

doi:10.2307/1969092

Cited by 24 publications

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“…In 1946, C. Nesbitt and R. M. Thrall showed [5] that if A is a Quasi-Frobenius algebra, then each faithful representation J? of A is equal to its own second commutator algebra R n \ In 1948 Thrall [6] initiated the study of the class of algebras A for which R = R" for each faithful representation R oί A.…”

Section: Denis Ragan Floyd Satisfying Either Of the Other Two Conditimentioning

confidence: 99%

“…Introduction* Throughout this paper, an algebra will be a finitedimensional associative algebra with identity over an arbitrary field. All modules are finite dimensional over these fields.In 1946, C. Nesbitt and R. M. Thrall showed [5] that if A is a Quasi-Frobenius algebra, then each faithful representation J? of A is equal to its own second commutator algebra R…”

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“…In 1946, C. Nesbitt and R. M. Thrall showed [5] that if A is a Quasi-Frobenius algebra, then each faithful representation J? of A is equal to its own second commutator algebra R…”

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OnQF-1 algebras

Floyd¹

1968

Pacific J. Math.

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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...

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Section: Denis Ragan Floyd Satisfying Either Of the Other Two Conditimentioning

confidence: 99%

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OnQF-1 algebras

Floyd¹

1968

Pacific J. Math.

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“…Let m¿ be the sum of all minimal subideals of U (that is, rrt,-is the "sockel" of 1¿). It follows from the general theory of rings [l, p. 8, Theorem 1.6B] that tn¿ can be written as a direct sum of minimal subideals of 1,-, say (6) m,-= nta + • • • + m¿r (direct). I As a minimal ideal, rrtjy is space for an irreducible representation %Pu¡) of 21, j = l, • • • , r. Since © is a faithful representation of 21 we must have some dominant ideal I,,«,-) which is not annihilated by tn,-/, j = 1, • • • , r. This means that we can find an element X=X(1(jy) in I^yj such that tttiyX^O and such that e¿X=X (where e¿ is a generating idempotent for I,-).…”

Section: Theorem 2 a Qf-2 Algebra Is Qf If And Only If (I) Ao Primitmentioning

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“…Properties of QF algebras and rings have been treated by Nakayama, Nesbitt, and the author [2,3,5,6](x). Some of the most important properties of QF algebras do not characterize these algebras, but occur in more extensive classes.…”

Section: Introductionmentioning

confidence: 99%