Geodesic
The group of units of a free product of rings
Sbornik. Mathematics, Tome 62 (1989) no. 1, pp. 41-63 
V. N. Gerasimov. The group of units of a free product of rings. Sbornik. Mathematics, Tome 62 (1989) no. 1, pp. 41-63. http://geodesic.mathdoc.fr/item/SM_1989_62_1_a2/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The main theorem asserts that the multiplicative group of a free product of rings, all of which satisfy the condition $xy=1\Rightarrow yx=1$, with the amalgamated skew field $\Lambda$, is a free product of a certain family of its subgroups with an amalgamated subgroup $\Lambda\setminus\{0\}$. As an application a ring $R$ is indicated for which the group $\operatorname{GE}_n(R)$ is a nontrivial free factor of $\operatorname{GL}_n(R)$ ($n$ being any natural number greater than one). Bibliography: 12 titles.

[1] Gerasimov V. N., “O svobodnykh lineinykh rpynnax”, XVII Vsesoyuzn. algebr, konf., Ch. 1, Minsk, 1983, 52–53

[2] Gerasimov V. N., “O svobodnykh proizvedeniyakh assotsiativnykh algebr”, KhVIII Vsesoyuzn. algebr. konf., Ch. 1, Kishinev, 1985, 113

[3] Golubchik I. Z., Mikhalev A. V., “Izomorfizmy polnoi lineinoi gruppy nad assotsiativnym koltsom”, Vestn. MGU. Ser. 1. matem., mekh., 1983, no. 3, 61–72 | MR | Zbl

[4] Golubchik I. Z., Mikhalev A. V., “O gruppe elementarnykh matrits nad $P1$-koltsami”, Vestn. MGU. Ser. 1. Matem., mekh., 1984, no. 1, 78 | MR

[5] Zelmanov E. I., “Izomorfizmy lineinykh grupp nad assotsiativnym koltsom”, Sib. matem. zhurn., 26:4 (1985), 49–67 | MR

[6] Lindon R., Shupp P., Kombinatornaya teoriya grupp, Mir, M., 1980 | MR

[7] Suslin A. A., “O stroenii spetsialnoi lineinoi gruppy nad koltsom mnogochlenov”, Izv. AN SSSR. Ser. matem., 41:2 (1977), 235–252 | MR | Zbl

[8] Bergman G. M., “Modules over coproducts of rings”, Trans. Amer. Math. Soc., 200 (1974), 1–32 | DOI | MR | Zbl

[9] Bergman G. M., “Coproducts and Some Universal Ring Constructions”, Trans. Amer. Math. Soc., 200 (1974), 33–38 | DOI | MR

[10] Cohn P. M., “On the free product of associative rings”, Math. Z., 71 (1959), 380–398 | DOI | MR | Zbl

[11] James D., Waterhouse W., Weisfeiler B., “Abstract Homomorphisms of Algebraic Groups: Problems and Bibliography”, Commun. in Algebra, 9:1 (1981), 95–114 | DOI | MR | Zbl

[12] Passman D. S., The Algebraic Structure of Group Rings, Wiley, N.Y., 1977 | MR | Zbl