Ideals Which Memorize the Extended Centroid

Abstract. Let A be a semiprime 2-and 3-torsion free non-commutative associative algebra. We show that the Lie algebra Der(A) of (associative) derivations of A is strongly non-degenerate, which is a strong form of semiprimeness for Lie algebras, under some additional restrictions on the center of A. This result follows from a description of the quadratic annihilator of a general Lie algebra inside appropriate Lie overalgebras. Similar results are obtained for an associative algebra A with involution and the Lie algebra SDer(A) of involution preserving derivations of A.

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“…Examples of extensions where Ann L (Q) = 0 are the dense ones (see [3] for the definition of a dense extension and [14] for examples).…”

Section: The Resultsmentioning

confidence: 99%

Strongly non-degenerate Lie algebras

Perera

Molina²

2008

Proc. Amer. Math. Soc.

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“…The condition expressed above is just a rephrasing of the density condition introduced by Cabrera in [5]. Specifically, for any algebra L (not necessarily associative) over our commutative ring of scalars Φ, let M (L) be the subalgebra of End Φ (L) generated by the identity map together with the operators given by right and left multiplication by elements of L. In the case of a Lie algebra L, note that M (L) is nothing but the unitization of A(L).…”

Section: Multiplicatively Semiprime Lie Algebras and Dense Extensionsmentioning

confidence: 99%

“…More concretely, if L ⊆ Q are Lie algebras, we consider when the natural correspondence A(L) → A(Q) given by the change of domain is an algebra map. This amounts to requiring that the extension L ⊆ Q is dense in the sense of Cabrera (see [5]).…”

Section: Introductionmentioning

confidence: 99%

“…Recall that an algebra A (associative or not) is said to be multiplicatively semiprime if A and the multiplication algebra M (A) (generated by the right and left multiplication operators of A together with the identity) are both semiprime (see [6], [7], [3], [4] among others). Roughly, an extension A ⊆ B of algebras is dense if every non-zero element in M (B) remains non-zero when restricted to A. Cabrera proved in [5] that every essential ideal of a multiplicatively semiprime algebra is dense.…”

Section: Introductionmentioning

confidence: 99%

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Associative and Lie algebras of quotients

Domènech

Molina²

2008

Publicacions Matemàtiques

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In this paper we examine how the notion of algebra of quotients for Lie algebras ties up with the corresponding well-known concept in the associative case. Specifically, we completely characterize when a Lie algebra Q is an algebra of quotients of a Lie algebra L in terms of the associative algebras generated by the adjoint operators of L and Q respectively. In a converse direction, we also provide with new examples of algebras of quotients of Lie algebras and these come from associative algebras of quotients. In the course of our analysis, we make use of the notions of density and multiplicative semiprimeness to link our results with the maximal symmetric ring of quotients.

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“…Later, another approaches to these concepts have appeared in the literature (see [19, §3; 20, §32]). At this point, we draw attention to the fact that the dense ideals of semiprime algebras memorize the extended centroid [12]. The aim of this paper is to initiate the study of complementedly dense ideals, by determining the extended centroid and the central closure of such ideals in a semiprime context, as well as to discuss the decomposable algebras with respect to the π-and ε-closures.…”

Section: Introductionmentioning

confidence: 99%

Complementedly Dense Ideals: Decomposable Algebras

2013

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An ideal I of a (non-associative) algebra A is dense if the multiplication algebra of A acts faithfully on I, and is complementedly dense if it is a direct summand of a dense ideal. We prove that every complementedly dense ideal of a semiprime algebra is a semiprime algebra, and determine its central closure and its extended centroid. We also prove that a semiprime algebra is an essential subdirect product of prime algebras if and only if, its extended centroid is a direct product of fields. This result is applied to discuss decomposable algebras with respect to some familiar closures for ideals.

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Ideals Which Memorize the Extended Centroid

Abstract: Essential ideals of multiplicatively semiprime algebras are themselves multiplicatively semiprime algebras and memorize the extended centroid.

Cited by 9 publications

References 6 publications

Strongly non-degenerate Lie algebras

Strongly non-degenerate Lie algebras

Associative and Lie algebras of quotients

Complementedly Dense Ideals: Decomposable Algebras

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