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Zachary Mesyan

Publications and source records attributed to Zachary Mesyan.

29 records · Page 2Linked to original sources
arXiv2012

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.

math.RA↗
arXiv2012

Simple Lie algebras arising from Leavitt path algebras

For a field K and directed graph E, we analyze those elements of the Leavitt path algebra L_K(E) which lie in the commutator subspace [L_K(E), L_K(E)]. This analysis allows us to give easily computable necessary and sufficient conditions to determine which Lie algebras of the form [L_K(E), L_K(E)] are simple, when E is row-finite (i.e., has finite out-degree) and L_K(E) is simple.

math.RA↗
arXiv2011

Groups where free subgroups are abundant

Given an infinite topological group G and a cardinal k>0, we say that G is almost k-free if the set of k-tuples in G^k which freely generate free subgroups of G is dense in G^k. In this note we examine groups having this property and construct examples. For instance, we show that if G is a non-discrete Hausdorff topological group that contains a dense free subgroup of rank k>0, then G is almost k-free. A consequence of this is that for any infinite set X, the group of all permutations of X is almost 2^|X|-free. We also show that an infinite topological group is almost aleph_0-free if and only if it is almost n-free for each positive integer n. This generalizes the work of Dixon and Gartside-Knight.

math.GR↗
arXiv2010

Conjugation of injections by permutations

Let X be a countably infinite set, and let f, g, and h be any three injective self-maps of X, each having at least one infinite cycle. (For instance, this holds if f, g, and h are not bijections.) We show that there are permutations a and b of X such that h=afa^{-1}bgb^{-1} if and only if |X\Xf|+|X\Xg|=|X\Xh|. We also prove a version of this statement that holds for infinite sets X that are not necessarily countable. This generalizes results of Droste and Ore about permutations.

math.GR↗
arXiv2009

The ideals of an ideal extension

Given a unital associative ring S and a subring R, we say that S is an ideal (or Dorroh) extension of R if for some ideal I of S, S = R + I, where the sum is direct. In this note we investigate the ideal structure of an arbitrary ideal extension of an arbitrary ring R. In particular, we describe the Jacobson and upper nil radicals of such a ring, in terms of the Jacobson and upper nil radicals of R, and we determine when such a ring is prime and when it is semiprime. We also classify all the prime and maximal ideals of an ideal extension S of R, under certain assumptions on the ideal I. These are generalizations of earlier results in the literature.

math.RA↗
arXiv2008

On minimal extensions of rings

Given two rings $R \subseteq S$, $S$ is said to be a minimal ring extension of $R$ if $R$ is a maximal subring of $S$. In this article, we study minimal extensions of an arbitrary ring $R$, with particular focus on those possessing nonzero ideals that intersect $R$ trivially. We will also classify the minimal ring extensions of prime rings, generalizing results of Dobbs, Dobbs & Shapiro, and Ferrand & Olivier on commutative minimal extensions.

math.RA↗
arXiv2007

Generating self-map monoids of infinite sets

Let I be a countably infinite set, S = Sym(I) the group of permutations of I, and E = End(I) the monoid of self-maps of I. Given two subgroups G, G' of S, let us write G \approx_S G' if there exists a finite subset U of S such that the groups generated by G \cup U and G' \cup U are equal. Bergman and Shelah showed that the subgroups which are closed in the function topology on S fall into exactly four equivalence classes with respect to \approx_S. Letting \approx denote the obvious analog of \approx_S for submonoids of E, we prove an analogous result for a certain class of submonoids of E, from which the theorem for groups can be recovered. Along the way, we show that given two subgroups G, G' of S which are closed in the function topology on S, we have G \approx_S G' if and only if G \approx G' (as submonoids of E), and that cl_S (G) \approx cl_E (G) for every subgroup G of S (where cl_S (G) denotes the closure of G in the function topology in S and cl_E (G) its closure in the function topology in E).

math.LO↗
arXiv2006

Commutator rings

A ring is called a commutator ring if every element is a sum of additive commutators. In this paper we give examples of such rings. In particular, we show that given any ring R, a right R-module N, and a set X, End_R(\bigoplus_X N) and End_R(\prod_X N) are commutator rings if and only if either X is infinite or End_R(N) is itself a commutator ring. We also prove that over any ring, a matrix having trace zero can be expressed as a sum of two commutators.

math.RA↗
arXiv2006

Endomorphism rings generated using small numbers of elements

Let R be a ring, M a nonzero left R-module, X an infinite set, and E the endomorphism ring of the direct sum of copies of M indexed by X. Given two subrings S and S' of E, we will say that S is equivalent to S' if there exists a finite subset U of E such that the subring generated by S and U is equal to the subring generated by S' and U. We show that if M is simple and X is countable, then the subrings of E that are closed in the function topology and contain the diagonal subring of E (consisting of elements that take each copy of M to itself) fall into exactly two equivalence classes, with respect to the equivalence relation above. We also show that every countable subset of E is contained in a 2-generator subsemigroup of E.

math.RA↗
arXiv2004

Generating Endomorphism Rings of Infinite Direct Sums and Products of Modules

Let R be a ring, M a left R-module, I an infinite set, N either the direct sum or product of |I| copies of M, and E the endomorphism ring of N as a left R-module. In this note it is shown that E is not the union of a chain of |I| or fewer proper subrings, and also that given a generating set U for E as a ring, there exists a positive integer n such that every element of E is represented by a ring word of length at most n in elements of U.

math.RA↗