Equality of orthogonal transvection group and elementary orthogonal transvection group

TL;DR

This paper proves that the orthogonal transvection group and the elementary orthogonal transvection group are equal for orthogonal modules with hyperbolic summands when the module splits locally, extending Bass's definitions.

Contribution

It establishes the equality of these two groups in the local splitting case, clarifying their relationship in the context of orthogonal modules.

Findings

The groups are equal when the orthogonal module splits locally.

This equality holds for modules with hyperbolic direct summands.

The result extends Bass's definitions to a broader class of modules.

Abstract

H. Bass defined orthogonal transvection group of an orthogonal module and elementary orthogonal transvection group of an orthogonal module with a hyperbolic direct summand. We also have the notion of relative orthogonal transvection group and relative elementary orthogonal transvection group with respect to an ideal of the ring. According to the definition of Bass relative elementary orthogonal transvection group is a subgroup of relative orthogonal transvection group of an orthogonal module with hyperbolic direct summand. Here we show that these two groups are the same in the case when the orthogonal module splits locally.