Defining relations of generalized orthogonal groups over commutative local rings without unit (Q1603047)

Let \(R\) be an associative ring without unit. For \(\alpha,\beta\in R\), one can define \(\alpha\circ\beta=\alpha+\alpha\beta+\beta\). An element \(\alpha\in R\) is called quasi-invertible if there exists \(\alpha'\in R\) such that \(\alpha\circ\alpha'=\alpha'\circ\alpha=0\). The group of quasi-invertible matrices from the complete matrix ring \(M(n,R)\) is denoted by \(\text{GL}^\circ(n,R)\) and called the generalized linear group of degree \(n\) over \(R\). In such a way, one can also define the generalized orthogonal group \(O^\circ(n,R)\). The author gives the defining relations of the generalized orthogonal group over a commutative local ring without unit.