منابع مشابه
Stably Free Modules
Let R be a commutative ring. When an R-module has a particular module-theoretic property after direct summing it with a finite free module, it is said to have the property stably. For example, R-modules M and N are stably isomorphic if Rk ⊕M ∼= Rk ⊕N for some k ≥ 0. An R-module M is stably free if it is stably isomorphic to a free module: Rk ⊕M is free for some k. When M is finitely generated a...
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We prove that for any finite-dimensional ringR and n ≥ dimR+2, the group En(R[X]) acts transitively on Umn(R[X]). In particular, we obtain that for any finite-dimensional ringR, all finitely generated stably free modules over R[X] of rank > dimR are free. This result was only known for Noetherian rings. The proof we give is short, simple, and constructive.
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We consider the class of crossed products of noetherian domains with universal enveloping algebras of Lie algebras. For algebras from this class we give a sufficient condition for the existence of projective non-free modules. This class includes Weyl algebras and universal envelopings of Lie algebras, for which this question, known as noncommutative Serre’s problem, was extensively studied befo...
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We give some examples of stably free modules which are not self dual.
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The construction of free R-modules over a cartesian closed topological category X is detailed (where R is a ring object in X), and it is shown that the insertion of generators is an embedding. This result extends the well-known construction of free groups, and more generally of free algebras over a cartesian closed topological category.
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ژورنال
عنوان ژورنال: Mathematika
سال: 2018
ISSN: 0025-5793,2041-7942
DOI: 10.1112/s0025579318000050