2017/09/17 by Claud W. G. Dias, Claud W. G. Dias Jr, Dias, Claud W. G. +2
Mathematics · #16R10 #16R40 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Rings and Algebras (math.RA) #math.RA #msc:16R10 #msc:16R40
paper · pdf · doi:10.48550/arxiv.1709.05728
18 p
arxiv created 2017/09/17 · openalex publication_date 2017/09/17 · arxiv updated 2017/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a group generated by a set X. It is well known and easy to check that [g1, g2, … ,gn] = 1 for all gi ∈ G \iff [x1, x2, … , xn] =1 for all xi ∈ X. Let L be a Lie algebra generated by a set X. Then it is also well known and easy to check that [h1, h2, … , hn] = 0 for all hi ∈ L \iff [x1, x2, … ,xn] = 0 for all xi ∈ X. Now let A be a unital associative algebra generated by a set X. Then the assertion similar to the above does not hold: for n > 2, it is easy to find an algebra A with a generating set X such that [x1, x2, … ,xn] = 0 for all xi ∈ X but [a1, a2, … ,an] ≠ 0 for some ai ∈ A. However, we prove the following result. Let R be a unital associative and commutative ring such that (1)/(3) ∈ R. Let A be a unital associative R-algebra generated by a set X. Let X2 = \ x1 x2 | xi ∈ X \ be the set of all products of 2 elements of X. Then [a1, a2, … ,an] = 0 for all ai ∈ A \iff [y1, y2, … , yn] =0 for all yi ∈ X ∪ X2. Moreover, one can assume that in the commutator [y1, y2, … , yn] above y1, yn ∈ X.