Composition-Diamond Lemma for Modules

Chen, Yuqun; Chen, Yongshan; Zhong, Chanyan

doi:10.48550/arxiv.0804.0917

Cited by 2 publications

(4 citation statements)

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“…In this section, by applying the Composition-Diamond lemma for modules (see [17,18]), we show that every countably generated k X -module can be embedded into a cyclic k X -module, where |X| > 1.…”

Section: Modulesmentioning

confidence: 99%

“…}. By Shirshov algorithm, we may assume that T is a Gröbner-Shirshov basis in the free module Mod k X Y in the sense of the paper [17] with the ordering (28) on X * Y .…”

Section: Modulesmentioning

confidence: 99%

“…is also a Gröbner-Shirshov basis in the free module Mod k X Y, y with the same ordering (28) on X * (Y ∪ {y}) since there are no new compositions. By the Composition-Diamond lemma in [17], M can be embedded into k X M ′ which is a cyclic k X -module generated by y. Then there exist f (x), g(x) ∈ k[x] such that y 1 = f (x)y, y 2 = g(x)y.…”

Section: Modulesmentioning

confidence: 99%

“…We systematically use Gröbner-Shirshov bases theory for associative algebras, Lie algebras, associative Ω-algebras, associative differential algebras, modules, see [40,11,16,17].…”

Section: Introductionmentioning

confidence: 99%

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Gröbner–shirshov Bases and Embeddings of Algebras

Bokut

Chen

2010

Int. J. Algebra Comput.

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In this paper, by using Gröbner-Shirshov bases, we show that in the following classes, each (resp. countably generated) algebra can be embedded into a simple (resp. two-generated) algebra: associative differential algebras, associative Ω-algebras, associative λ-differential algebras. We show that in the following classes, each countably generated algebra over a countable field k can be embedded into a simple two-generated algebra: associative algebras, semigroups, Lie algebras, associative differential algebras, associative Ω-algebras, associative λ-differential algebras. Also we prove that any countably generated module over a free associative algebra k X can be embedded into a cyclic k X -module, where |X| > 1. We give another proofs of the well known theorems: each countably generated group (resp. associative algebra, semigroup, Lie algebra) can be embedded into a two-generated group (resp. associative algebra, semigroup, Lie algebra).

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Section: Modulesmentioning

confidence: 99%

“…}. By Shirshov algorithm, we may assume that T is a Gröbner-Shirshov basis in the free module Mod k X Y in the sense of the paper [17] with the ordering (28) on X * Y .…”

Section: Modulesmentioning

confidence: 99%

Section: Modulesmentioning

confidence: 99%

“…We systematically use Gröbner-Shirshov bases theory for associative algebras, Lie algebras, associative Ω-algebras, associative differential algebras, modules, see [40,11,16,17].…”

Section: Introductionmentioning

confidence: 99%

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