Injectively homogeneous rings

Brown, Kenneth A.; Hajarnavis, C. R.

doi:10.1016/0022-4049(88)90078-3

Cited by 14 publications

(29 citation statements)

References 12 publications

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“…It is verified that universal enveloping algebras of finite-dimensional Lie algebras over fields of prime characteristic are injectively homogeneous in the sense of [BH88]. As a consequence of the general theory of injectively homogeneous rings developed in [BH88] we obtain a nice description of a minimal injective resolution of the universal enveloping algebra as a module over itself in terms of the injective hulls of its prime factor rings considered as one-sided modules. In particular, this enables us to show that the last term of such a minimal injective resolution is isomorphic to the continuous dual which was proved by Barou and Malliavin [BM85] for finite-dimensional solvable Lie algebras over algebraically fields of characteristic zero.…”

Section: Introductionmentioning

confidence: 79%

Injective Modules and Prime Ideals of Universal Enveloping Algebras

Feldvoss¹

2006

Abelian Groups, Rings, Modules, and Homological Algebra

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In this paper we study injective modules over universal enveloping algebras of finite-dimensional Lie algebras over fields of arbitrary characteristic. Most of our results are dealing with fields of prime characteristic but we also elaborate on some of their analogues for solvable Lie algebras over fields of characteristic zero. It turns out that analogous results in both cases are often quite similar and resemble those familiar from commutative ring theory.

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Section: Introductionmentioning

confidence: 79%

Injective Modules and Prime Ideals of Universal Enveloping Algebras

Feldvoss¹

2006

Abelian Groups, Rings, Modules, and Homological Algebra

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“…The definitions of homological and injective homogeneity (for a ring R which is a finite module over its centre) are recalled from [10,11] in Definition 5.1; in essence, these conditions require that simple R-modules with the same central annihilator share certain properties in common. The relation of the hom.hom.…”

Section: Homological Hierarchymentioning

confidence: 99%

“…An error which occurs in both [10] and [11], concerning the dependence of these homogenity conditions on the choice of central subring, is corrected in Remarks 2.14(iii) and (iv).…”

Section: Then the Following Are Equivalent: (I) R Is Injectively Homomentioning

confidence: 99%

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The Cohen Macaulay Property for Noncommutative Rings

Brown

Macleod

2017

Algebr Represent Theor

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Let R be a noetherian ring which is a finite module over its centre Z(R). This paper studies the consequences for R of the hypothesis that it is a maximal Cohen Macaulay Z(R)-module. A number of new results are proved, for example projectivity over regular commutative subrings and the direct sum decomposition into equicodimensional rings in the affine case, and old results are corrected or improved. The additional hypothesis of homological grade symmetry is proposed as the appropriate extra lever needed to extend the classical commutative homological hierarchy to this setting, and results are proved in support of this proposal. Some speculations are made in the final section about how to extend the definition of the Cohen-Macaulay property beyond those rings which are finite over their centres.

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“…A Noetherian ring S which is a finite module over a central subring C is centrally Macaulay if S is a Cohen-Macaulay C-module. The key property we require concerning such rings is stated in the next lemma, which, though implicit in [2], [3], [4], is not stated explicitly there. (ii) Let c be a regular element of S. Since S is a finite Z-module, O^cSnZ.…”

Section: Sufficient Conditions In the Abelian-by-finite Casementioning

confidence: 99%