Gröbner-Shirshov bases and embeddings of algebras

Details Gröbner-Shirshov bases and embeddings of algebras Gröbner-Shirshov bases and embeddings of algebras

2009-08-13

arxiv.org/abs/0908.1992v1

International Journal of Algebra and Computation, 20(2010), 875-900

Mathematics

Rings and Algebras

26 pages

Scientific paper

In this paper, by using Gr\"obner-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 $\Omega$-algebras, associative $\lambda$-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 $\Omega$-algebras, associative $\lambda$-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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