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## Scientific Bulletin. Physical and Mathematical Research

#### Abstract

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 .

#### First Page

70

#### Last Page

76

#### References

1. Šemrl, P. (1997) Local automorphisms and derivations on B(H), Proceedings of the American Mathematical Society. Vol. 125. Issue 9. pp. 2677-2680. 2. Kim, S., Kim, J. (2004) Local automorphisms and derivations on Mn, Proceedings of the American Mathematical Society. Vol. 132. Issue 5. pp. 1389-1392. 3. Lin, Y., Wong, T. (2006). A note on 2-local maps, Proceedings of the Edinburgh Mathematical Society. 49(03). pp 701-708. 4. Ayupov, Sh., Kudaybergenov, K. (2012) 2-local derivations and automorphisms on B(H). Journal of Mathematical Analysis and Applications. Vol. 395. Issue 1. pp. 15-18. 5. Ayupov, Sh., Arzikulov, F. (2014) 2-Local derivations on semi-ﬁnite von Neumann algebras. Glasgow Mathematical Journal. 56(1). pp 9-12. 6. Ayupov, Sh., Kudaybergenov, K. (2015) 2-Local derivations on von Neumann algebras. Positivity 19(3). pp 445-455. 7. Arzikulov, F. (2012). Infinite norm decompositions of C*-algebras. Operator Theory: Advances and Applications. Vol. 220. pp 11-21. 8. Ayupov, Sh., Arzikulov, F. (2017). 2-Local derivations on associative and Jordan matrix rings over commutative rings. Linear Algebra and its Applications. 522. pp 28-50. 9. Ayupov, Sh. A., Arzikulov, F.N. (2018). Description of 2-local and local derivations on some Lie rings of skew-adjoint matrices. Journal of Linear and Multilinear Algebra. October. DOI: 10.1080/03081087.2018.1517719 10. Ayupov, Sh. A., Arzikulov, F.N. (2017) 2-Local Derivations on AW*-Algebras of Type I. Lobachevskii Journal of Mathematics. 38. Issue 1. pp 148-161. 11. Ayupov, Sh. A., Arzikulov, F.N. (2018). 2-Local derivations on algebras of matrix-valued functions on a compact. Vladikavkaz Mathematical journal. Vol. 20. Issue 1. pp. 38-49.

#### Recommended Citation

Arzikulov, F. N.; Umrzakov, N. M.; Nuriddinov, O. O.; and I. S. Zaynobiddinov
(2019)
"DESCRIPTION OF 2-LOCAL DERIVATIONS ON AN ALGEBRA OF MATRICES,"
*Scientific Bulletin. Physical and Mathematical Research*: Vol. 1
:
Iss.
1
, Article 9.

Available at:
https://uzjournals.edu.uz/adu/vol1/iss1/9