arXiv subjects

Cao Minh Nam

Publications and source records attributed to Cao Minh Nam.

3 recordsLinked to original sources

arXiv2026

Orthogonality of Defect Ideals for Jordan Homomorphisms

Let $φ\colon A\to B$ be a Jordan homomorphism of associative algebras over a field $\mathbb{F}$ of characteristic different from $2$, with defect ideals $V_φ$ and $W_φ$. If $A=\operatorname{Alg}([[A,A],[A,A]])$, then $V_φW_φ=W_φV_φ=0$ and $V_φ\cap W_φ\subseteq\operatorname{Ann}(\operatorname{Alg}(φ(A)))$. Moreover, the induced map modulo $V_φ\cap W_φ$ is standard. The annihilator bound is sharp, even in finite dimension. As an application, when $A$ is unital and $A=K(A)$, every Jordan homomorphism defined on $A$ is splittable and standard. In this setting, the corresponding result of Brešar and Zelmanov holds without the surjectivity hypothesis.

math.RA↗

arXiv2026

Locally Solvable Radicals via Wilson Radical Sets and Subgroup Lattices

For a group $G$, let $S(G)$ be the set of elements $g \in G$ such that $\langle g,x\rangle$ is solvable for all $x \in G$. We study when $S(G)$ coincides with the locally solvable radical $R_{\mathrm{L}\mathfrak{S}}(G)$. Using Wilson's profinitely convergent word sequences, we show that, for every locally (solvable-by-finite) group $G$ and every Wilson sequence $ω$, $$ R_{\mathrm{L}\mathfrak{S}}(G) = R_{\mathrm{L}\mathfrak{R}}(G) = S(G) = W_ω(G).$$ We also obtain four-conjugate and seven-commutator descriptions of radical membership, together with a two-conjugate result for torsion elements of order coprime to $6$. The same radical identity holds for locally linear groups, and hence for subgroups of $\mathrm{GL}_{\infty}(D)$ when $D$ is a locally finite-dimensional division ring. Independently, we prove that groups with nearly modular subgroup lattice satisfy $$R_{\mathrm{L}\mathfrak{S}}(G) = R_{\mathrm{L}\mathfrak{R}}(G) = S(G).$$

math.GR↗

arXiv2020

On locally finite skew group algebras

In this note, we study the problem on the existence of non-cyclic free subgroups of the skew group algebra of a locally finite group over a field.

math.RA↗