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In the article from the same positions group homomorphisms of matrix groups and ring homomorphisms of matrix rings over associative rings from 1 are described. It is shown that the description of homomorphisms of matrix groups E (n, R) ⊆ G ⊆ GL(n, R), n ≥ 2 into the group of automorphisms GL(W) of the left (optionally free) K-module W over an arbitrary associative ring K of 1 is reduced to cases where 2 or 3 are reversible elements in the ring K. It is proved that they allow a standard description of homomorphisms of the group of elementary transvections $E(n,R), if such a description allows homomorphisms of matrix groups over rings K, in which 2 or 3 are reversible elements. The ring homomorphisms are also described Λ : Rn → End W, n ≥ 2 of the left (optionally free) K-module W over an arbitrary associative ring K of 1. It is shown that homomorphisms Λ allow a standard description on the ring Rn.

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).

The paper substantiates the image by formal matrices of some elements of linear groups over associative rings by using the properties of redundant and fixed modules. The form of images of elements of linear groups with respect to their homomorphisms in the group of automorphisms of modules over associative rings of 1, in which there are reversible elements 2 or 3, is shown. Using this approach, the authors described homomorphisms with the condition (*) of matrix groups over associative rings with 1. In particular, the isomorphisms of matrix groups over associative rings are homomorphisms with condition (*).

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)., Вивчення гомоморфiзмiв матричних груп над асоцiативними кiльцями розпочалося майже 100 рокiв тому роботами Шраєра i Ван-дер-Вардена i в подальшому розвивалися в працях Дьєдоне, Хуа Ло-гена, Райнера, О’Мiри, Хана, Ю.I. Мерзлякова, Уотерхауса, О.В. Мiхальова, Ю.I. Зельманова, I.З. Голубчика, В.М. Петечука та iнших авторiв. В основi вивчення знаходяться груповi властивостi повної лiнiйної групи GL(n, R) – множини всiх оборотних матриць над асоцiативним кiльцем R з 1. При n ≥ 3 у всiх вiдомих випадках, незважаючи на вiдмiннiсть методiв, якi застосовувалися, автоморфiзми повної лiнiйної групи виявлялись добутком стандартних автоморфiзмiв. Саме оборотнiсть елемента 2 давала можливiсть розглядати все бiльш широкi класи кiлець над якими можливий стандартний опис гомоморфiзмiв матричних груп. Якщо 2 – необоротний елемент, то при n ≥ 3 В.М. Петечук зробив опис автоморфiзмiв групи GL(n, R) у випадку, коли R – комутативне локальне кiльце. Виявилося, що при n ≥ 4 всi автоморфiзми таких груп є добутком стандартних автоморфiзмiв, а при n = 3 їх можна виразити через стандартнi i деякий нестандартний автоморфiзми. Спираючись на цей результат, В.М. Петечук [2] отримав опис iзоморфiзмiв групи GL(n, R), n ≥ 3, якщо R – довiльне комутативне кiльце. Зокрема, вiн здiйснив опис гомоморфiзмiв Λ : P E (n, R) → P GL(m, K), m ≥ 3, n ≥ 3 таких, що ΛP E (n, R) = P H i H ⊇ E (m, K) над довiльними комутативними кiльцями R i K. I.З. Голубчик i О.В. Мiхальов [3], використовуючи системи iдемпотентiв, i незалежно Ю.I. Зельманов [4], використовуючи методи йорданових алгебр, отримали опис iзоморфiзмiв групи E (n, R), n ≥ 3, 2 ∈ R∗ на групу E (m, K), 2 ∈ K∗ над довiльними асоцiативними кiльцями R i K з 1. В.М. Петечук [5] зробив опис гомоморфiзмiв групи P E (n, R), n ≥ 3 в групу GL(m, K), m ≥ 2, 2 ∈ K∗ у випадку, коли нерухомi пiдмодулi деяких елементiв четвертого порядку збiгаються з нерухомими пiдмодулями їх квадратiв. З нього випливають результати I.З. Голубчика, О.В. Мiхальова i Ю.I. Зельманова. Розвиваючи технiку, пов’язану з iдемпотентами, I.З. Голубчик [6] здiйснив опис iзоморфiзмiв груп GL(n, R) i GL(m, K) при n, m ≥ 4 над асоцiативними кiльцями R i K. Виявилося, що вони допускають стандартний опис на групi E (n, R). Авторами В.М. Петечук, Ю.В. Петечук [7, 8] описанi гомоморфiзми з умовою (*) з чого зокрема випливає i опис iзоморфiзмiв повних лiнiйних груп над асоцiативними кiльцями. У данiй роботi удосконалюються i розширюються методи опису гомоморфiзмiв з умовою (*), якщо елемент 2 є оборотним в кiльцi K i n ≥ 3. Основним результатом роботи є наступна теорема. Нехай R i K – асоцiативнi кiльця з 1, 2 ∈ K∗ , E (n, R) ⊆ G ⊆ GL(n, R), n ≥ 3, W – лiвий K-модуль, гомоморфiзм Λ : G → GL(W) задовольняє умову (*). Тодi Λ має стандартний опис на групi E (n, R).

The paper substantiates the image by formal matrices of some elements of linear groups over associative rings by using the properties of redundant and fixed modules. The form of images of elements of linear groups with respect to their homomorphisms in the group of automorphisms of modules over associative rings of 1, in which there are reversible elements 2 or 3, is shown.Using this approach, the authors described homomorphisms with the condition (*) of matrix groups over associative rings with 1. In particular, the isomorphisms of matrix groups over associative rings are homomorphisms with condition (*)., В роботi обґрунтовується зображення формальними матрицями деяких елементiв лiнiйних груп над асоцiативними кiльцями шляхом використання властивостей лишкових i нерухомих модулiв. Показано вигляд образiв елементiв лiнiйних груп вiдносно їх гомоморфiзмiв у групу автоморфiзмiв модулiв над асоцiативними кiльцями з 1, в яких елементи 2 або 3 є оборотними.За допомогою даного пiдходу авторами були описанi гомоморфiзми з умовою (*)матричних груп над асоцiативними кiльцями з 1. Зокрема, iзоморфiзми матричних груп над асоцiативними кiльцями є гомоморфiзмами з умовою (*).