Group generated by elementary matrices over a unital ring

Definition
Let $$R$$ be a unital ring and $$n$$ be a natural number. The group generated by elementary matrices of degree $$n$$ over $$R$$, denoted $$E_n(R)$$, is defined as the subgroup of the general linear group $$GL_n(R)$$ generated by the elementary matrices $$e_{ij}(\lambda)$$ for $$1 \le i,j \le n$$, $$i \ne j$$, and $$\lambda \in R$$. The elementary matrix $$e_{ij}(\lambda)$$ is a matrix with $$1$$s on the diagonal, $$\lambda$$ in the $$(ij)^{th}$$ position, and $$0$$s elsewhere.

There is a natural homomorphism from the Steinberg group over a unital ring $$St_n(R)$$ to this group. For a commutative unital ring and also for a division ring, we can define a determinant homomorphism and a special linear group $$SL_n(R)$$, and $$E_n(R)$$ is a subgroup of the special linear group. For a field, division ring, or Euclidean domain, $$E_n(R)$$ equals $$SL_n(R)$$.