The present paper is devoted to 2-local derivation on associative and Jordan matrix rings. 2-local derivation is defined as follows: given a ring , a map (not additive in general) is called 2-local derivation if for every , there exists a derivation such that and . In 1997, P. Semrl introduced the notion of 2-local derivations and described 2-local derivations in the algebra ¬ of all bounded linear operators on the infinite-dimensional separable Hilbert space . A similar description for the finite-dimensional case appeared later in 2004. In the paper Y. Lin and T. Wong 2-local derivations have been described on matrix algebras over finite dimensional division rings. In 2012 Sh. Ayupov, K. Kudaybergenov suggested a new technique and have generalized the above mentioned results for abritrary Hilbert spaces. Namely they considered 2-local derivations on the algebra of all linear bounded operators on an arbitrary (no separability is assumed) Hilbert space and proved that every 2-local derivation on is a derivation. Later Sh. Ayupov, K. Kudaybergenov and F. Arzikulov extended the above results and gave the proof of the theorem for arbitrary von Neumann algebras. A number of papers were devoted to 2-local maps on diﬀerent types of rings, algebras, Banach algebras and Banach spaces. The on 2-local derivations on ﬁnite dimensional Lie algebras, on weak-2-local derivations, 2-local ∗-Lie isomorphisms and 2-local Lie isomorphisms were obtained as a result. It proved a number of theorems on 2-local triple derivations. Other classes of 2-local maps on diﬀerent types of associative and Jordan algebras were also studied. A concept of 2-local left multiplication on an arbitrary ring is introduced and studied. This notion is introduced as follows: let be a ring. Then a mapping of into itself is called 2-local left multiplication if for every elements there exists an element such that , . In the present paper, it is proved that every 2-local left multiplication of some associative rings is a left multiplication. Namely, we prove that for any element there exists an element such that . Besides, in the present paper, a concept of 2-local Jordan multuplication on an arbitrary Jordan ring it is introduced and studied. This notion is introduced as follows: let be a Jordan ring. Then a mapping of into itself is called 2-local Jordan multiplication if for every elements there exists an element such that , . Finally, it is proven that every 2-local Jordan multiplication of some Jordan rings is a Jordan multiplication. Namely, it is proven that for every element there exists an element such that .
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Arzikulov, F. N.; Umrzakov, N. M.; Nuriddinov, O. O.; and I. S. Zaynobiddinov
"DESCRIPTION OF 2-LOCAL DERIVATIONS ON AN ALGEBRA OF MATRICES,"
Scientific Bulletin. Physical and Mathematical Research: Vol. 1
, Article 9.
Available at: https://uzjournals.edu.uz/adu/vol1/iss1/9