Group generated by elementary matrices over a unital ring

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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 1s on the diagonal, \lambda in the (ij)^{th} position, and 0s 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).