Gröbner-Shirshov bases and embeddings of algebras

Abstract

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 $\Omega$-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).

26 pages

References in corpus (3)

• Groebner-Shirshov Bases for Lie Algebras: after A. I. Shirshov
• Composition-Diamond lemma for differential algebras
• Composition-Diamond Lemma for Modules

Cited by in corpus (1)

• Filtrations and Distortion in Infinite-Dimensional Algebras