[Paper Review] 2-Local derivations on matrix rings over associative rings

Shavkat Ayupov, Farhodjon Arzikulov|arXiv (Cornell University)|Mar 25, 2013

Advanced Topics in Algebra6 references3 citations

TL;DR

This paper establishes that every inner 2-local derivation on the matrix ring $M_n(\Re)$ over a commutative associative ring $\Re$ is an inner derivation, and that every derivation on $\Re$ extends to a derivation on $M_n(\Re)$. The results generalize prior findings on 2-local derivations to commutative matrix rings and provide a foundational extension theorem for derivations in non-unital and non-separable settings.

ABSTRACT

In the present paper it is proved that every inner 2-local derivation on the matrix ring $M_n(\Re)$ of $n imes n$ matrices over a commutative associative ring $\Re$ is an inner derivation. Also, it is proved that, every derivation on an associative ring $\Re$ has an extension to a derivation on the matrix ring $M_n(\Re)$ of $n imes n$ matrices over $\Re$.

Motivation & Objective

• To determine whether inner 2-local derivations on matrix rings over commutative associative rings are inner derivations.
• To investigate the extension of derivations from an associative ring $\Re$ to the matrix ring $M_n(\Re)$.
• To generalize existing results on 2-local derivations from finite-dimensional and Hilbert space settings to general commutative associative rings.
• To establish conditions under which 2-local derivations on associative rings are additive and hence derivations.
• To explore the structural properties of inner 2-local derivations in non-unital and non-separable algebraic settings.

Proposed method

• Define inner 2-local derivations as maps $\Delta: M_n(\Re) \to M_n(\Re)$ such that for any $x,y$, there exists $a \in M_n(\Re)$ with $\Delta(x) = [a,x]$ and $\Delta(y) = [a,y]$.
• Use matrix units $\{e_{i,j}\}$ to decompose elements and analyze the action of $\Delta$ on elementary matrices.
• Prove that the action of $\Delta$ on $e_{i,j}$ and $\sum e_{k,k+1}$ implies consistency in the associated commutators via lemmas on matrix component identities.
• Construct an extension of derivations from $\Re$ to $M_n(\Re)$ by first extending to $M_2(\Re)$, then to $M_4(\Re)$, and so on, using a projection-based restriction to $M_n(\Re)$.
• Apply induction and subring extension techniques to show that derivations on $\bar{M}_2(\Re) \cong M_2(\Re)$ extend to $M_n(\Re)$.
• Use the fact that if $\Re$ is generated by two elements and $\Delta$ is additive, then $\Delta$ is an inner derivation via commutator structure on polynomials in generators.

Experimental results

Research questions

• RQ1 Is every inner 2-local derivation on $M_n(\Re)$, where $\Re$ is a commutative associative ring, necessarily an inner derivation?
• RQ2 Can every derivation on an associative ring $\Re$ be extended to a derivation on $M_n(\Re)$?
• RQ3 Under what conditions is a 2-local derivation on an associative ring $\Re$ additive and hence a derivation?
• RQ4 Does the extension of derivations from $\Re$ to $M_n(\Re)$ hold for non-separable or infinite-dimensional settings?
• RQ5 What structural constraints ensure that an inner 2-local derivation on a ring generated by two elements is an inner derivation?

Key findings

• Every inner 2-local derivation on $M_n(\Re)$ for a commutative associative ring $\Re$ is an inner derivation.
• Every derivation on an associative ring $\Re$ extends to a derivation on $M_n(\Re)$, as shown via step-by-step extension through $M_2(\Re)$, $M_4(\Re)$, etc.
• The extension of derivations is achieved by restricting a larger derivation on $M_{2^k}(\Re)$ to $M_n(\Re)$ using a projection $e = \sum_{i=1}^n e_{i,i}$.
• For rings $\Re$ generated by two elements, if an inner 2-local derivation $\Delta$ is additive, then $\Delta$ is an inner derivation.
• The result fails for $n=1$, as shown by a counterexample in $U_2(\mathbb{C})$, where an inner 2-local derivation is not additive.
• The lattice of projections in a von Neumann algebra being non-atomic implies the existence of 2-local derivations on $S(\mathcal{M})$ that are not derivations, indicating limitations in extension properties.