Yet Another Variation on the Theme of Decomposition of Transvections

The unipotent decomposition method consists in representing elementary matrices as products of factors belonging to proper parabolic subgroups whose images under endomorphisms (e.g., conjugations) remain in proper parabolic subgroup. For the complete linear group, this method was suggested in 1987 by Stepanov, who applied it to simplify the proof of Souslin's normality theorem. Soon after this, Vavilov and Plotkin transferred the method to other classical groups and the Chevalley groups. Since then, many results in the same spirit have been obtained. The paper suggests yet another variation on this theme. Namely, let R be a commutative ring with identity, and let g is an element of GL(n, R), where n >= 4. Then, the elementary group E(n, R) is generated by transvections e + uv, where u is an element of R-n, v is an element of R-n, and vu = 0, such that v, gu, and vg(-1) have at least one zero component each. This result is related to a simplified proof of theorems of Waterhouse, Golubchik, Mikhalev, Zel'manov, and Petechuk about the auto-morphisms of the complete linear group being standard, which uses unipotent elements.