Varieties of algebras

Osborn, J. Marshall

doi:10.1016/0001-8708(72)90003-5

Cited by 95 publications

(51 citation statements)

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“…Note that by our assumption 8 EFq. If var F 2j 3 A's, then as we have already noticed at the end of §4 of this paper, A's is a subdirect product of fields F t. But A'6 is a simple algebra (see [6,Proposition 15.4]), so A's -F f, which is impossible since A'8 has nontrivial idempotents. Thus,F 2sE var A's and by Proposition 9, Lemma 2 from Theorem 12, we have 2s|2, i.e.…”

Section: Proofmentioning

confidence: 84%

“…Thus, x is algebraic, dimfe F < °° and since 7(F) = 0, F=0"_ xFqd¡ and therefore, var F = var F r for some F r from this decomposition. By Jacobson's theorem (see [6], [11]) two-generated algebras in V are commutative and hence, V is associative (see [12,Lemma 3]), since F has an identity X = XqT. Lemma 1.…”

Section: Proofmentioning

confidence: 99%

“…Let V be a variety of (7, hyalgebras over a field F , where p = char Fq > 5 for arbitrary (7,5) Note that if k is an infinite field, char k ¥= 2, 3 then (i)-(ü) are the only varieties of (-1, l)-algebras with a chain of subvarieties, since in this case they are homogeneous. 6. Jordan algebras.…”

Section: Proofmentioning

confidence: 99%

“…By Proposition 7 the matrix T from equation (7) Proof. Let B E V. By Theorems 15.11, 15.4 from [6] and Proposition 6, B is a subdirect product of fields F t, t\r, and algebras As over some fields F ,. By Proposition 6 we assume 8 EFq.…”

Section: Proofmentioning

confidence: 99%

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On Chain Varieties of Linear Algebras

Артамонов¹

1976

Transactions of the American Mathematical Society

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ABSTRACT. In the present paper we study varieties of linear k-algebras over a commutative associative Noetherian ring k with 1, whose subvarieties form a chain. We describe these varieties in terms of identities in the following cases: residually nilpotent varieties, varieties of alternative, lordan and (-1, l)-algebras.1. Introduction. In [1] we have already described homogeneous chain varieties of linear algebras over a field. RecaU that variety is homogeneous iff its free algebras are graded relative to the degree-function. If the ground field is infinite, then any variety of algebras over this field is homogeneous. In the second paper [2] we gave the description of chain varieties of restricted p-algebras Lie over an infinite field.In the present paper we describe in Theorems 2-15 chain varieties V of linear fc-algebras, fc-commutative associative Noetherian ring with 1, in the following cases:(i) if G is a free algebra in V then f\nGn = 0 (this is always vaUd when V is locaUy nüpotent), i.e. V is residuaUy nilpotent, (ü) F is a variety of alternative algebras, (iii) V is a variety of (-1, 1 )-algebras, (iv) F is a variety of Jordan algebras. Throughout the paper k is a fixed commutative associative ring with 1, and V is a variety of fc-algebras with chain of subvarieties. We also assume that

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Section: Proofmentioning

confidence: 84%

Section: Proofmentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

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On Chain Varieties of Linear Algebras

Артамонов¹

1976

Transactions of the American Mathematical Society

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“…We also rely heavily on the ability to linearize identities [3], [5] and on the linear independence of various elements of our algebra. Thus, we restrict our attention to algebras over fields.…”

Section: Introductionmentioning

confidence: 99%