Special linear group over a commutative unital ring

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Definition

Let R be a commutative unital ring and n be a natural number. The special linear group of degree n over R, denoted SL_n(R) or SL(n,R), is defined as the subgroup of the general linear group comprising those matrices whose determinant is 1. Equivalently, it is the kernel of the determinant homomorphism.

Here, the determinant for a matrix is defined in the usual way as a polynomial function. Note that this function is independent of the choice of basis, hence the special linear group can be considered more abstractly for any free module over R without an explicit basis.

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