Гомоморфiзми з умовою (*), якщо 2 – оборотний елемент

Jazyk: angličtina
Rok vydání: 2020
Předmět:
Zdroj: Науковий вісник Ужгородського університету. Серія: Математика і інформатика, Vol 2, Iss 37, Pp 101-113 (2020)
ISSN: 2616-7700
Popis: The study of homomorphisms of matrix groups over associative rings began almost 100 years ago with the work of Schreier and Van der Warden and later developed in the works of Dieudonne J., Hua L. K., Reiner I., O’Meara O.T., Hahn A.J., Merzlyakov Yu.I., Waterhouse W.C., Mikhalev O.V., Zelmanov E.I., Golubchik I.Z., Petechuk V.M. and other authors. The study is based on the group properties of a complete linear group GL(n, R) the set of all reversible matrices over the associative ring R with 1. Thus, in all known cases n ≥ 3, despite the difference in the methods used, the automorphisms of the complete linear group were the product of standard automorphisms. It is the reversibility of element 2 that made it possible to consider ever wider classes of rings over which a standard description of homomorphisms of matrix groups is possible. If 2 is an irreversible element, then when n ≥ 3 Petechuk V.M. described the automorphisms of the group GL(n, R) in the case when R is a commutative local ring. It turned out that for n ≥ 4 all automorphisms of such groups are the product of standard automorphisms, and for n = 3 them can be expressed through standard and some non-standard automorphisms. Based on this result, Petechuk V.M. [2] obtained a description of the isomorphisms of the group GL(n, R), n ≥ 3 if R is an arbitrary commutative ring. In particular, he described homomorphisms Λ : P E (n, R) → P GL(m, K), m ≥ 3, n ≥ 3 such ΛP E (n, R) = P H and H ⊇ E (m, K) as over arbitrary commutative rings R and K. From Golubchik I.Z. and Mikhalev O.V. [3], using systems of idempotent, and independently Zelmanov E.I. [4], using the methods of Jordan algebras, obtained a description of the isomorphisms of the group E (n, R), n ≥ 3, 2 ∈ R∗ on the group E (m, K), 2 ∈ K∗ over arbitrary associative rings R and K with 1. Petechuk V.M. [5] described the homomorphisms of a group P E (n, R), n ≥ 3, into a group GL(m, K), m ≥ 2, 2 ∈ K∗ in the case when the fixed submodules of some elements of the fourth order coincide with the fixed submodules of their squares From it follow the results of Golubchik I.Z., Mikhalev О.В. and Zelmanov E.I. Developing techniques related to idempotents, Golubchik I.Z. [6] described the isomorphisms of the groups GL(n, R) and GL(m, K) for n, m ≥ 4 over the associative rings R and K. It turned out that they allow a standard description on the group E (n, R). The authors Petechuk V.M., Yu.V. Petechuk [7, 8] described homomorphisms with condition (*), from which in particular follows the description of isomorphisms of complete linear groups over associative rings. In this paper, methods for describing homomorphisms with the condition (*) are improved and expanded if element 2 is reversible in the ring K and n ≥ 3. The main result of the work is the following theorem. Let R and K be associative rings with 1, 2 ∈ K∗ , E (n, R) ⊆ G ⊆ GL(n, R), n ≥ 3, W be the left Kmodule, homomorphism Λ : G → GL(W) satisfies the condition (*). Then Λ has a standard description on the group E (n, R).
Databáze: OpenAIRE
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