Special linear group over a commutative unital ring

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.

Related notions

 * Special linear group over a field is the typical use.
 * Special linear group over a unital ring: A generalization based on a corresponding generalization of the notion of determinant to the non-commutative case.
 * Special linear group over a division ring
 * Group generated by elementary matrices over a unital ring. This is contained in the special linear group, but need not be the whole special linear group. When the ring is a Euclidean domain or a division ring, it coincides with the whole group.
 * Steinberg group over a unital ring: This has a natural homomorphism surjecting to the group generated by elementary matrices. The homomorphism is an isomorphism in the case of fields.