66 research outputs found
Commutator Leavitt path algebras
For any field K and directed graph E, we completely describe the elements of
the Leavitt path algebra L_K(E) which lie in the commutator subspace
[L_K(E),L_K(E)]. We then use this result to classify all Leavitt path algebras
L_K(E) that satisfy L_K(E)=[L_K(E),L_K(E)]. We also show that these Leavitt
path algebras have the additional (unusual) property that all their Lie ideals
are (ring-theoretic) ideals, and construct examples of such rings with various
ideal structures.Comment: 24 page
Growth, entropy and commutativity of algebras satisfying prescribed relations
In 1964, Golod and Shafarevich found that, provided that the number of
relations of each degree satisfy some bounds, there exist infinitely
dimensional algebras satisfying the relations. These algebras are called
Golod-Shafarevich algebras. This paper provides bounds for the growth function
on images of Golod-Shafarevich algebras based upon the number of defining
relations. This extends results from [32], [33]. Lower bounds of growth for
constructed algebras are also obtained, permitting the construction of algebras
with various growth functions of various entropies. In particular, the paper
answers a question by Drensky [7] by constructing algebras with subexponential
growth satisfying given relations, under mild assumption on the number of
generating relations of each degree. Examples of nil algebras with neither
polynomial nor exponential growth over uncountable fields are also constructed,
answering a question by Zelmanov [40].
Recently, several open questions concerning the commutativity of algebras
satisfying a prescribed number of defining relations have arisen from the study
of noncommutative singularities. Additionally, this paper solves one such
question, posed by Donovan and Wemyss in [8].Comment: arXiv admin note: text overlap with arXiv:1207.650
Finite higher commutators in associative rings
If is any finite higher commutator in a ring , for example , and if has minimal cardinality so that the ideal generated by is infinite, then is in the center of and . Also, if is any finite, higher commutator containing no nonzero nilpotent element then generates a finite ideal.
DOI:
10.1017/S000497271300089
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